I was writing out a post for the modpage, but as it was getting very long I thought it might be better here. I've been playing this with LWotC, which is worth keeping in mind for the balance feedback I'll give below.

I'd echo what a few people seem to have said - very fun, wildly overpowered, for a few reasons;

Too flexible - the class can do pretty much anything, which only gets worse as you level up. It can tank for most of your team (early game I don't even bother with cover), it has a rocket launcher, dash melee for one action which shreds, light em up and an extremely potent self heal if the unit ever does take meaningful damage.
I'm around the end of midgame at the moment with the T3 melee weapon (meaning I also set enemies on fire) and I've noticed the rocket scales to add even more damage and rupture. Dread to think what the T3 handblaster will do. I took Total Combat from the Xcom skill row, meaning I also have single action grenades and a free grenade slot (though I'm not sure whether that's part of the mod out of the box).
I see in the Perks and Points thread that there's an emergency teleport ability. Is this still a thing? It's definitely not required - this guy has never come close to getting killed and I'm not exactly careful with him. This implies I could have been even more reckless.

Overpowered rankup abilities - A 1 turn cooldown single action teleport, a single-action guaranteed perma-haywire mech takeover with 3 charges, better Overdrive allowing 3 actions in one turn even if they're melee dash actions (it's insane how much you can get done with this), secondary melee attack to disable weapons, electro psi-lance (another devastating skill when combined with overdrive)... honestly, it says a lot that I went with the massive AoE cluster missile at the last rank and I almost never use it.

No weakness - I suspect the low mobility was the intended downside of the class. However, the teleport skill is more than enough to move around faster than other soldiers, and you can use the overdrive + melee combo to get around it low mobility too (3 movement + melee attack for one action). The only other downside I've found is I can't use command on the Proxian, but with its insane action economy I don't really need to. As a robot, the Proxian also has fire, bleed, psi, KO, crushing and poison immunity.

A few suggestions:

Hyperdrive should be last rank ability. It doesn't work well with the missile barrage (a two action ability) so it makes sense to have them on the same tier, and it's far too powerful to be obtained so early. This makes many of the Proxian's other abilities far more balanced until top rank, and creates more of a trade-off.

The rocket launcher should not be built in - make this a perk instead. The melee, durability and light-em-up abilities are good enough at the low ranks.

Nerf or remove teleport - it's just too good at circumventing weaknesses, and I don't know why anyone would pick anything else at the tier. The longer cooldown shockwave teleport might be safe to keep; I haven't tried it because all you need out of a teleport ability is the movement.

Nerf Nano Construct - even with one charge, this is a crazy ability due to guaranteed success. I'm going to be genuinely happy when I run into sectopods with this. It should have one charge, and some success chance based on enemy hack defence, since the perma-control is so powerful. It perhaps should be a two action ability - it certainly should if it has a guarantee on success and works with overdrive.

Nerf to raw stats - it should have the same health as a regular solider at best if it's going to have built in armour - it will still be durable, but you may actually have to consider defensive small items instead of ignoring them. There's also no reason for it to have aim and hack stats superior to most/all other soldiers. In a pinch, I can teleport to the objective. hack it and have a very good chance of getting a reward, on top of everything else the Proxian does.