I had a perm a couple years ago, and the stylist was concerned that my waves would take so much curl from even the mildest perming solution on the biggest curling rods available, that I'd end up looking like Ronald McDonald.

So you'd have to talk to a stylist about it (preferably one who has a LOT of experience with perms), but it's unlikely you'd be able to get a looser curl on such curly hair from a regular perm, since the perm rods used are so small.

There are varying degrees of relaxers available. Not that I"d necessarily recommend them, since they're just as damaging to the hair as perms (probably even more damaging), but if you really can't stand your natural curl pattern, that might be worth looking into.
co-wash - Suave Naturals Coconut
condish - Trader Joe's TTT and Refresh
styling - FOTE, shea butter, jojoba oil, LA Looks

Kryptonite: Stearalkonium Chloride, quats