No problem.
A FA FCG WILL fit into a semi auto restricted lower as well. Everything minus the auto sear pin and spring obviously.
You just end up being able to fire a single round on FA selector and then you get a malfunctioned gun ie "dead mans trigger" with a live round in the chamber and have to rack the action, expend the still live cartridge to load another one. Then you can fire that one and repeat.
FA FCG are not technically prohibited and can be installed in a semi auto lower. It's the auto sear and having it installed which is the prohibited arrangement. Case in point, one dealer was bringing in complete USGI FA FCG included in complete A1 parts kits for legal sale not long back.
A full auto capable lower will have a longer recess cut out of the far rear of the receiver and a hole bored through the receiver just above the selector to mount the auto sear.
That receiver would be a FA receiver and prohibited as such.
A semi auto receiver has no auto sear hole and the rear of the receiver is left much more shallow.
As long as the lower you bought is a semi auto lower with "auto" simply inscribed on the selector area, you will be fine Sir.