If my opponent used Magnemite's "Mirror Shot" attack last turn, and I try to use my GX attack this turn and flip tails, do I lose the ability to use a GX attack for the rest of the game?
No, because since you flipped tails the attack "doesn't happen".
Source: Cosmic Eclipse FAQ (2019-10-18), TPCi Rules Team (2019-10-31)

Relevant cards

Note: All applicable cards may not be listed below.
COL1_5
DPP_DP32
DPP_DP32
HGSS3_3