I can’t believe this was staring me in the face the whole time…
The secret, if you’re looking to get the FPL app on your phone for free is to simply go to https://m.fantasy.premierleague.com.
Then, just bookmark/add the page to your Home screen and, hey presto, you have as close to the official app as you can get, without paying for it!
As far as I know this should work for both iOS and Android devices.
Let me know if this tip is helpful or if you’ve found a better app for FPL.