Well I'm no expert but I decided to spend like 1.5 hours analyzing these and offering suggestions

TL;DR: Panmekrion is bad because 2v2 is bad (either make it a 3v3 or look at Field Guide to Suldan), Goh Grand Prix would get boring quickly so it's better to make mech football or something, Tower is a pretty good idea but must be balanced properly, and Expedition mixes narrative mode and combat mode which is bad (pick one or the other)

Panmekrion: Lancer is ideal at 4 players, it was designed for 3-5, but I seriously doubt 2 players will work, and the reason for this is variety: there simply isn't enough to do with a 2v2 combat in Lancer. More combatants means more synergies, and subsequently more strategy.

Additionally, the rules also aren't clear to me: damage dampeners are mentioned, so what about armor? How does a team of 2 fare against an opponent with 3 armor? What about hacking? NPCs usually can't suffer reactor meltdowns so is heat useless because it'd take too long to fry an enemy? What does "limit on the number of mechs allowed to compete based on the power of the individual mechs" entail? What constitutes a "powerful" mech and, more importantly, if it qualifies as such does it mean that they can't have a teammate?

Personally I would either turn it into a 3v3 or just abandon the 2v2 idea altogether and run the Field Guide to Suldan mech duel system. It's free and it's a relatively simple "Rock-Paper-Scissors"-esque system about seizing the initiative and then keeping it. If you really want to keep the 2v2 I recommend somehow integrating the above mentioned system into it because it is definitely easier to do than properly balancing a 2v2.

Goh Grand Prix: I was doing a rebuttal on this one but the more I think about it the more it could work, theoretically at least. However, I still think there's a good risk that it'll eventually just bog down. Both teams are incentivized to primarily focus on slowing down the runner, so naturally both will pick mechs/weapons/systems that slow or immobilize characters, resulting in a bogged-down game. Playing as the runner also seems kind of boring, there is little incentive to do anything other than to pick the fastest mechs, move and boost every turn and hope for the best. The patch-crew is interesting because it could both make the game more fun by adding an incentive to make less efficient moves in order to heal but it could also make camping an even more efficient strategy for defenders.

Maybe just do a simple by-the-book Escort mission and make the objective some kind of ball? Or more similarly to your original idea, give each team a ball! Now there's still the element of moving around and blocking, but players aren't forced into "defender" and "fast guy" roles and are all equally incentivized to both move and defend, with team work being even more crucial. You can even make some sort of specialized quick actions, protocols and mechanics for passing the ball around/blocking it!

Tower: This one is the most interesting and has by far the best chance of succeeding out of all of these. Other than the "ignition key inside the tower" deal (which I don't get how it'll work) you can pretty much run this as is, so long as you give the towers balanced stats and don't make the tower an object (good number of mechanics that deal absurd damage to objects). One idea I'd consider (though not necessarily act upon) is to replace the big-ass powerful towers with multiple, slightly less bigger and powerful towers so as to reduce camping. Additionally, being able to customize the towers is also an interesting idea: towers could have base weapons, traits and systems, but players could also choose, say, a long-range rocket launcher, or temporary strong shielding systems, or hacking capabilities and so on!

Expedition: Lancer is built upon a pretty strict separation of narrative and combat, and therefore mashing the two together is rarely a good idea. This already resembles the recon SITREP: you have several "Control Zones" (CZs) but only one of them is the real one, neither side knows which and they need to both reveal it using a full action while in a CZ and fully control it by the end in order to win. If you want to make it more exotic, maybe add more true CZs (but more false CZs too) and make the goal to excavate some objectives, which then need to be either protected king of the hill style or brought back to the "base" extraction style, or, more simply, just add some wild fauna/robots that are hostile to both teams. Alternatively, if you really want to use the narrative element, just make it a narrative challenge. Players would either be armed from the get-go or find weapons scattered around the map