This is sort of where I'm going with this but I was trying not to duplicate the same "type" of unit.
8 of my 10 lances are named already:
Assault
Command
Fire
Heavy
Medium
Pursuit
Recon
Striker
The remaining 2 are those above that I'm looking for alternatives to.
You could use the Support name for another "Fire" lance.
Cavalry for a fast flexible lance (literally meant to plug your holes that open)
I've always like "Battle" or "Trooper" for a basic fighting lance.
Light is open if you have that. It might be a slower light lance. Think things like Valks, Panthers etc.
Other way to go is dispense with the type designations for the last two. Those are "Named" Make them particularly flashy, dynamic. Maybe even mismatched. A mech unlikely to be seen in that faction. Maybe an Age of War model, or a Star League refit.
Suggestions
Bryant's Brawlers
Curtiss' Cuthroats
Langston's Lancers
Paul's Pillager Lance
Don't have to be all rhymey and full off alliteration. (Repeating the consonant sound) or assonance (repeating the vowel sound)
Call them "The Thugs"
"Puppy Huggles" with whacky color schemes and a penchant for kicking in cockpits. Commanded by a loyal but utterly psychopathic relative of the battalion commander.
Have fun and don't sweat it too much.