I also have similar arrangements on bench/reserve players' conditions before such matches. Any player comes off the bench have slight disadvantage in condition. I generally accept this because that's not a serious problem and fair to all players. But it should be nice if there is a simple solution for this.
Assume match period is 12 minutes, the probability is about 12 / (60x3) = 1/15. So it's almost 2 league games per season.