The quickest way is simple just play the team of the week team in squad battles and quit the game once the ball has kicked off. When you hit 200 game just ride it out and wait for it to unlock during the game I'm sure.
This is how it's been for years.
Helpful to rack up a bunch of contracts before you begin this.
*** Disclaimer ***
After a lengthy number of quitting squad battle games you may encounter an error which ends up preventing you from doing so. Fear not, the "temp ban" should be over in a day and just play a few games fully then attempt this again.
This solution is to help you save time from this grindy achievement if you for whatever reason don't like the FUT game mode and want the achievement over. It's simple enough to complete as we all know but the time saver certainly helps.
Alternative method (credit to Neo Yggdrasill in the comments)
Find someone who already has a player with 200+ appearances and do a co-op squad battle match. Should you have zero luck with that, the original solution would suffice for progress much quicker than the legit way.