Padmé is your best bet for seeing improvements in Squad Arena.
Your best 2 toons for immediate benefit, however, are Geo Spy and Vader. Spy requires some of the same gear as Padmé (she requires a stun gun and Spy requires 2 right now)
As a result, I would make Padmé your #1 priority and Vader #2 since they don't need conflicting gear. Padmé needs one Mk5 Stun Gun at g11, but Vader doesn't need any. Nor does JKA. Ackbar & Thrawn also have to be priorities. So here is my near-term gear priority list for you:
R2 => g8+5
CLS => g7 or g8 if g8 seems easy to get
Padmé => g12
Vader => g12
Tarkin => g10
Ackbar => g11
Spy => g12
Anakin => g12
Thrawn => g11
Ackbar => g12
R2 should be 7* now. You qualify for the 7* version of his event, and even without Palp's zeta your AoE stun should be able to win the battle for you. Now, you won't get a good result with the AoE stun every time, but you can replay the event over and over, so just try a few times and you should be able to promote R2 to 7* today - or this week, at least.
Moreover, once you have R2 at 7* and g8, you should have CLS. Everything else you need for CLS is already in place except for the Omega on Farmboy's first special ("Bullseye"). Give farm boy that one Omega and r2 g8/7* and you have CLS. (Okay, you MIGHT need to get Old Ben to g8, but I think he can do it at g7 where you have him if you just pay some cash to switch mods around. Give him the fastest speed possible, with 2nd consideration being Tenacity and 3rd consideration being survivability - Health, Protection & defense.)
I would still let CLS languish on gear for a bit though. g7 is enough for now since you don't yet have all the other toons that make him fearsome as a leader. Also because he won't come into his own without zetas -- at least 2 -- which probably aren't worth it until his whole team can make use of them.
Remember that Padmé and Vader are using different gear to get to g12 than R2 and CLS are using to get to their goals. You should be able to work on R2, Padmé and Vader all at once, and CLS, Padmé & Vader at once after unlocking CLS.
I would also try subbing Padmé's squad for your Arena offense at least a couple times per day. See if that goes better than Geos, and if it does, make the switch permanent and start looking at how you can shift your best mods away from Geos and onto your Padmé squad.
As for zetas, however, I still think that Palp's leadership is the next priority. Then Vader's Merciless Massacre. Then you can start worrying about JKA & GMY (who deserve a zeta each) and CLS. Yes, your empire squad isn't exactly ready for prime time, but that one Palp zeta makes more of a difference than any other zeta you could add right now. Conquest is sexy and I know you want to do better, but there are other game modes as well, and having an extra squad is always important. Even in Conquest. (You can, for example, use Padmé or Geos to get ahead one more node in Conquest then use Palp to go back to Sector 1 to work on feats while Padmé's stamina refreshes - to get the most out of conquest, you always need a bare minimum of 2 squads since it takes only 3 hours to get enough energy to try another battle, but 5 hours to recover the stamina from your last one). But even 2 good squads isn't enough since specific teams require specific counters. Since a Palp-led team has very different strengths and weaknesses from a Padmé or Geo squad, I like bringing it along as quickly as you can. That's also part of why I have Tarkin & Thrawn on your gear priorities list.
One big concern I have is that you don't have General Kenobi unlocked yet. You should be getting more than 10 shards per week of GK right now, since you're with a guild that should be simming HAAT. Is your guild not using the Sim function? Your Padmé squad won't reach its potential without GK.
Now, even with all this set out, you're still lacking a 5th Empire or Sith toon to go with Palp. I see your Bastila (Fallen) is only 2*. Are there any shards you've saved up but not used to promote her? She is an excellent 5th and IIRC uses few or no Mk5 stun guns.
So where is Bastila in your farming plans, and what Empire toons are you farming right now. You need to figure out a plan for that 5th toon now so you know what direction you're headed.
Re: Fleet:
You've still got a bank of GW tokens to use only on Sun Fac, right? His ship will be much more useful at 7* than he is at 5* where you have him now. And I trust you're still putting GW Tokens into finishing GeoSoldier's Starfighter as well.