If you want to use launching uppercut then you are going to be readying actions, and you want to be good at bull rushing. The raging bull boxing talent = rhino charge, take whichever of those you like so you can ready a charge. Also you'll be getting improved & greater bull rush. Combat reflexes may be useful given the AoOs you hope to take.
You're going to be wanting some mobility, so take the athletics sphere and maybe a talent or two there as well as your free spheres. Remember you get a martial tradition too.
Also pummeling charge requires BAB +12 or 8th level brawler - you can get it already and you probably want to. Which mostly rules out other styles. You can still get other styles via martial flex when you want of course.
Extra combat talent has the [combat] tag so yes, martial flex works on it. You can even get multiple copies of it at once.
I can only assume that a lot of people were using pummeling charge in PFS and the people running it deduced from that the feat was too good.
I thought you were starting at level 10? Anyway, 5 levels: 3 feats (4 if human), boxing and open hand spheres free, 1 martial tradition, 3 talents.
Power attack is obvious, as is pummeling style, then you might get improved bull rush with the remaining feat. If human consider combat reflexes.
If you get the pit fighter martial tradition (just guessing, but it sounds like the right background given your description?) then you get a bonus boxing sphere talent, the gladiator sphere, and another talent from one of those.
You said you wanted axe kick and launching uppercut so get those, that leaves 1 general talent plus the bonus talents from pit fighter. Raging bull and overhead smash synergise with launching uppercut, as would chasing assault if you're allowed legendary talents. If not you might pick up a boast talent from the gladiator sphere - whatever appeals.
Brutal pummeler trades in the brawler bonus feats, you don't get those. You can't take pummeling style until you get the brawler's flurry at L2 though that's a minor detail, just switch the order of those feats a little.
From brutal pummeler you get boxing & open hand.
From pit fighter you get gladiator & a talent from boxing or gladiator, and the double-up on the boxing sphere converts to another boxing sphere talent.
Then you get 3 talents from being L5 with the adept progression.
Total: boxing, open hand & gladiator spheres; one boxing talent, one talent from boxing or gladiator, three general talents.
Launching uppercut looks fine to build around. I'm not saying it's the best thing ever, or even in the top twenty for PF melee, but it's a useful trick.
It lets you make a standard action attack then if that hits, you can make a trip attempt at -2 as an immediate action, then if that succeeds you can make two AoOs with the bonus for the enemy being prone (one with 2*Str mod to damage, one with 1*Str mod I think.)
Not bad if it all comes together, but if you miss with either of the first two rolls you don't get the AoOs. The odds just aren't that good. I'm fairly sure you could put together a better way to trip the enemy and so get the AoOs.
Another means of tripping, or something based around launching uppercut? Should I assume legendary talents are OK or not?
Any normal melee attack (i.e. not another combat maneuver, or ranged attack, or touch spell delivery or similar) can be replaced by a trip attempt. Any successful trip will trigger those AoOs if you have axe kick, snap kick & dual opportunity. Dual attack from the dual wielding sphere lets you make an attack with your main hand and your off-hand as a standard action.
So, first thing in the round you take a move action: this could be to move up to the enemy, to use martial flex to gain a feat, you could even use the base ability of the open hand sphere to make an attempt to trip at -2.
Next (assuming you weren't in position to use the base ability of open hand, or that it didn't succeed), do that standard action dual attack. The first attack is a trip attempt. If it succeeds, take your double AoO, then do a normal attack with the off hand. If it fails, try to trip with the off hand, then if that succeeds do your AoOs. If both fail you're out of luck.
This gives you the two AoOs if either trip attempt works, rather than needing both a hit and a successful trip.
Also it requires one less combat talent.
Does that make sense?
You're getting 3 talents in open hand there and you have 3 general talents to get them. You need either a martial tradition which gives open hand (which will convert to a talent in open hand) or one which gives dual wielding to get all those and the dual wielding sphere too. Weapon master would be the easy answer but there may be one which fits your concept better.
A brutal pummeler trades in their unarmed damage progression as well so they wouldn't get that bonus. Just the listed damage by number of talents on that table.
You'll want the giantslayer and maybe titan breaker feats from spheres of might so that trip (or bull rush if you go back to launching uppercut) works and becomes easier on big enemies.
Pummeling charge like I said before. It's a no-brainer for an unarmed attacker, especially one who can get it early.
You may want the fighter feats weapon focus, weapon spec, greater weapon focus, greater weapon spec. I'm not sure you'll have the feats free of course given that you lose the brawler bonus feats but taking weapon focus so you can flex the rest is a possibility.
Talent-wise there are several dual wielding talents which give extra attacks, and even if you're not going to use launching uppercut or other counters you may like some boxing talents like shoulder roll or cross counter. If your GM at all likes setting up fights where you'll need to move significantly look at athletics sphere options.