I also have the same problem. When I use "Move to Club" feature to get a player into my club (Arsenal), they are often ineligible to use as it says "Player cant play for more than 2 teams in same season" or "The player is involved in other match today already".
Example:
I transfered player "Igor Kalashnikov" from Lokomotiv Moscow to Arsenal, he kept playing for Lokomotiv Moscow aswell as being availavable to me SOMETIMES (!) due to previously mention problem. When I click on his profile in FMRTE, it says he's an Arsenal player, unique ID and all but when I go check team panels of Arsenal and Lokomotiv Moscow, he's on both panels and when you click on either of players in team screens, both link to 1 screen and that is the one of my Arsenal version of the player.
Basically, using "Move to Club" makes a copy of the player and transfers it to destination, leaving 1 buggy copy of the player in the old club and playing for the old club, making the "new copy" that is in the new club, practically unusable.