I don't know the origins of your game, nor the CFW mode, so here's some info of mine, if it makes a difference.
Shield, Cart, on Non-CFW device.
Sword, Cart, on Atmosphere CFW device.
Both share the same user profile. Both able to trade. Both on same firmware and same game version.
Based on my experience (and what I've read), CFW won't make the device "not able to trade".
I've read something on GBAtemp about if both device share the same user profile, unable to trade, but I'm assuming that is related to digital versions of the game. Since mine is both physical version, as well as differing version, I don't seem to be affected by that. As far as I can, since yours is also differing versions, I doubt it's that.
Do you happen to have incognito installed on the CFW version?
edit:
Just create a new user profile. No need to delete the current save.
New user profile = new save. For all we know, it's the old user profile that was the problem.
Creating a new user profile hits 2 birds with one stone.