The OP says that the FLEX was playing defense every other inning. Remember, the FLEX is a position in the line-up. A team can go back and forth between using, or not using, the FLEX an unlimited number of times in a game.
Now, if you're talking about the player occupying the FLEX position that is a different story. The actual player is subject to the same re-entry rules that any other player occupying any other position is subject to.
How about this:
Sally is the DP. Mary is the FLEX.
inning 1: Mary/FLEX plays defense. All good so far.
Inning 2: Sally/DP plays defense. Mary/FLEX sits out and has left the game.
Inning 3: Mary/FLEX plays defense again. This requires one re-entry.
Inning 4: Sally/DP plays defense again. Mary/FLEX sits out. She is now out of the game permanently.
Inning 5: Now, we have an issue.
At what point was it discussed about the FLEX playing every other inning? As this shows, it could be the fifth inning before it even becomes an issue with the starting FLEX (Mary).
Of course, they're free to put a sub in for Mary and the sub would become the new FLEX player.
For Inning 5, the new player occupying the FLEX position (let's call her Cindy) plays defense.
Inning 6: Sally/DP plays defense. Cindy/FLEX sits out and has left the game. She has one re-entry left.
Inning 7: Cindy/FLEX re-enters and plays defense.
It is possible for the FLEX to play every other inning for an entire game. The same player can't be the FLEX in this case. Maybe they were trying to use the same player from the fifth inning on, but that's not what the first post said.