Entirely new types are possible, but there is a good variety with what the current ones cover. I think the best way to handle that without stepping on other exos toes is to add them in as variations the current exos, maybe only showing up once you upgrade with the bahumet device. So current exos would have the designation "prime" such as Assault Prime, and then new ones could be designated "Omega" or some such.
Examples
Brawler: forgoes even medium ranged weaponry, disentigrator replaced with flamethrower (splash damage in a line from target square to the exo), pistol replaced by PUNCH (range 1, range cannot be increased, very high base damage and enough ammo to be practically unlimited). every third (or maybe even every second) movement does not use an action, with a counter on the action bar keeping track.
Assault: likely just different weapon selection, Heavy Machine Gun instead of laser rifle + light machine gun (like chaingun but instead of a free shot every even shot it has a free shot whenever it kills a bot), micro grenades instead of grenade launcher (smaller splash/damage more range/ammo), that kind of thing.
Sniper: smaller sight bonus but can see/sensor through cover like its not there (maybe even doors/false walls?), alternate gamma laser that can fire through cover like its not there but only gets a free shot when ammo is divisible by 3. Shadow torpedo's instead of plasma rifle (or maybe instead of rail gun).
Ninja: losses ability to attack without breaking stealth, but gets a large damage bonus when attacking while stealthed.
Science: half the built in bonus to computer related skills, starts with virus points instead of hacking points, can hack/virus from range (lets say range 3). we can roll ops commander thoughts into this and let it virus friendlies (turrets/virused bots) to add some bonus stats.
Siege: likely just Different weapons, like a rocket launcher that spawns a rocket that has a limited range per turn, but persists turn to turn (wont alert bots, and already alert bots have a high chance (like 95%+) to ignore it. essentially giving it an unlimited range, and able to target around corners, but takes time to reach the target. Almost Like shadow torpedoes with splash, a time delay, and unable to go through walls/doors.
You stuff like that.