I rambled a bit below, as I often do -- just ignore the parts where I explain something obvious
This particular sort of deck is tricky to build in KS. Without
Seasoned Leader,
Elite Archer and
Southron Bowman are really unimpressive bearers of the
Mumak. And without
Ships of Great Draught it's harder to maximize
Arrow from the South once it's loaded. Still, as with everything it's got potential in the format!
Broad observation is you don't have two parts of your deck, you have four -- archery, skirmishing (mumaks), threats, and twilight. I think any two parts can go together quite well, but the more you add the more intentional every card has to be. Archers are the main spot, with Mumaks complementing them with
Arrow from the South (and of course wounds). Since there's no
Rallying Call you have to really devote yourself to threats in KS, it's not simply part of every Southron deck.
War Towers can be efficient enough on its own to justify as a bonus, but I don't think you're generating them consistently enough for
Mumakil of the Harad to be a real value. Do you get at least
back from it? If not,
Flanking Attack or
Trample will probably be better. Which brings us to the twilight portion. Right now just 4 cards need an excess of twilight, so I'd imagine that more often than not gaining twilight doesn't actually do much (well, except get a stop and maybe bluff having
Whirling Strike or
Red Wrath. Which is not nothing, but not going to get you any kills). The mumaks are ideally net 0 twilight, so they don't bring you any closer to playing anything in your current setup.
Arrow from the South is a card I quite like, but have struggled to find a good use for. In my latest attempt I just get 2-3 tokens and use it after that, often only on
Desert Lord. The note you make about exhausting but not killing might have to do with how your deck is split -- you've got 12 archers to exhaust but then only 3 cards to kill when the FP may not expect it,
Whirling Strike and
Red Wrath. The high skirmish strength is certainly a factor, but easier for the Free Peoples to manage since they know exactly what they're gonna get. If there's a 3rd string fighter who can take a big guy on with a pump, you probably won't see a return on your work until site 8 or 9. This is where
Arrow from the South shines to me, putting the wounds back on the main fighters at site 7 after the FP spent the sanctuary healing them. In such a scenario,
Southron Captain's damage +2 isn't going to matter much, the FP will just throw him on an exhausted companion.
Here's the goal I see for this deck: you make it so the main fighters are all one bad assignment away from a sudden death, which forces second string to fight and lose skirmishes which racks up even more wounds. Eventually this forces the FP to a breaking point and their whole fellowship comes crashing down. The archer,
mumak, and twilight parts all play very well into this goal, even if the twilight part is a little wedged in at the moment. Threats, however, do not -- Frodo is the fighter of last resort, and depending on The Ring he might be the only one to ever take those wounds. No wounds are
bad, but some are worse than others and these don't contribute to what the rest of your deck is doing.
As far as what I'd do... Load the deck up with a bunch of rares
Apparently all my Southron decks are extremely rare heavy and I hadn't noticed. Just about every card I would suggest is R, and half the second string was too.
Southron Fighter,
Desert Stalker,
War Mumak, and
Southron Troop are the biggest ones I'd add if you have 'em.
Ithilien Wilderness and
Wrath of Harad are good too, put a pump out of your hand.
Ambush will let you gain some twilight from fierce skirmishes instead of just breaking even, great to tie together the
mumak and twilight parts.
Southron Commander with a
Mumak will pretty quickly trim an opponent's FP down to a manageable level, and your archer text will make sure they can't just discard exhausted nobodies. 13 strength is no slouch too. Just be wary of extra copies of main fighters which might allow them to come back fully healed.
Southron Invaders' text probably won't come into play but he's also a reasonably competent minion with ambush. Instead of
Mumakil of the Harad or the other generators I mentioned, I'd use
Raider Halberd. The exerts are a valid concern for your archers but your non-archers are the ideal
Mumak bearers and they've got no use for their vitality. Plus, it's enough for your opponent that you
could gain twilight for
Whirling Strike or
Red Wrath when assigning minions, which may help to bluff better. It's also enough to combo with
Army of Haradrim on its own, exhaust him for 6 twilight and play a
mumak.
Raider Bow only nets you 3!
I've always been really unimpressed by
Mumak Commander, for the reasons you note.
Southron Captain is basically the same in a broader set of circumstances for cheaper. I don't know what you're using for site 5,
City Gates* is great the cap'n but if you don't get him by then it can sometimes benefit the FP by removing threats (if they're less than 3 away from the threat limit).
Pelennor Flat* can benefit
Discovered, and if you go that route I'd consider
Desert Legion. A bid of 2 will lead opponents to a normally safe 4 burdens, turning him into a fearsome foe. You wouldn't be able to use
City Gates to clear threats for your own FP though, and the hand discard / burden choice can bite you just the same. It's a real shame there's only 2 options for site 5 in KS.
So, for a deck that's focused on archery and skirmishing with a bit of twilight thrown in, here's how I would use your sideboard:
-2x
Mumak Commander (too expensive)
-1x
Bold Men and Grim (removing the threat part of the deck)
-3x
War Towers (ditto. It's a fine card, but threat wounds will go on RB and undermine your archers' efforts to ignore his vitality)
-2x
Mumakil of the Harad (obviously)
+1x
Army of Haradrim (solid minion with good text)
+1x
Southron Traveler (great ambush even with middling strength)
+1x
Ulaire Enquea, Lieutenant of Morgul (because I removed
War Towers and
Mumak Commander... I'm assuming you have him and don't have
Southron Commander. You can consider replacing another supporting card for
War Towers to add back threats for
Morgulduin if you often expect to see 6+ companions, or use another copy of Traveler instead of him if you rarely expect to see 6)
+3x
Raider Halberd (Good for the
mumak and twilight portions simultaneously)
+1x
Whirling Strike (expanding the twilight part of the deck)
+1x
New Strength Came Now (expanding twilight part in a different direction)
I might swap a
Discovered or even a
Mumak for another minion, depending on how things go. If you wind up not seeing the value out of
Arrow from the South that you'd like, reducing it to 1x to be a fun bonus rather than a particular aim for the deck is also a solid idea. I've got you up a minion from before even though I really wanted to see 2
NSCN. My thinking is 3-4 skirmishes and an event in hand is going to be better than 2-3 skirmishes and 2 events. Nearly took out
Southron Captain for
Southron Wanderer, but even if he's not as helpful as other (rare) minions might be a 13 strength potential-mumak-bearer is pretty solid for this deck. Plus he's a great
Raider Halberd candidate.
Southron Wanderer is not my favorite minion anyway, he's a fine generator early on but once Frodo has a sword the potential mind games probably won't give good enough return in this build where you're pretty clearly not trying to wound the RB (and, between you and me, won't be able to actually overwhelm him without obviously overextending). I decided to leave your second
Red Wrath out in hopes that the archers will all do their job well enough that you're never 2 wounds away from killing a skirmishing character, and that your opponent will generally put the exhausted archers on their fighters. Some would disagree that you should ever put
Whirling Strike above
Red Wrath though, depends on how you want to play.
The hope is that your deck is more tuned to killing the strongest FP characters. Exhaust them with archers, finish 'em with an event, then just plow through whatever weaklings are left with the additional skirmish wins that this brings. Ambush minions will add assignments without costing as much twilight, which'll more often put you in the position to off someone. If you can score a kill or two at site 7 by unloading
Arrow from the South into
Whirling Strike, that's probably the ideal. Site 8 should get you 2 more after successfully doing that (one planned via assignment, one forced via event; or both to a big
mumak) and then you're set to skirmish the Ring-bearer at 9. Still a bit worried about the minion count. Big fellowships make this plan pretty hard to realize, the path there is 1) get 5 wounds out of Enquea and 2) use all that excess twilight up with
Army of Haradrim. Not very clean I admit, but it should at least slow your opponent considerably.
Southron Commander would turn that situation around on his own though.