There is nothing that says it doesn't work the way you described it (with no proficiency and disadvantage for a weapon that's a bit heavy for it's size). It couldn't use a heavy crossbow at all but, the light one with disadvantage is fine.
According to DMG p.278 (optional rules):
A creature has disadvantage on attack rolls with a weapon that is
sized for a larger attacker. You can rule that a weapon sized for an
attacker two or more sizes larger is too big for the creature to use
There is no actual rule RAW that prohibits tiny creatures from using any weapons, but the only tiny creature I found using weapons is a sprite and it does 1 damage per attack that hit's with its longsword and its shortbow(+ poisoned), which probably means they are tiny weapons since they don't even have a damage die.
A small person has disadvantage when using heavy crossbows and a tiny person can carry half the amount according to PHB p. 176. So going by logic it wouldn't be able to even wield it, but by RAW it could use it since it can lift it with it's Strength of 6(Can Lift 6/2*30=90 lbs and carry 45 lbs). But i wouldn't allow it to load bolts.
Light crossbow is over 3 times lighter so it shouldn't be as difficult to use for the Imp.