I'm hoping to be able to remove the coin doubler I bought a long time ago, but I don't want to lose my progress and downloaded characters. Is this possible?
You don't, and you can't.
Well, not without using save editors anyway...
You may be asking...
This is because of the way saves are implemented. On both iOS and Android, saves are tied to each device the game is played on, and that the status of whether or not a save has a coin doubler (like Jetpack Joyride) is in the save file.
No, because these features are all in-game purchases.
So if you decide to restore your purchases, your coin doubler and downloaded chars will be lost. Otherwise your progress will not be impacted because it's stored on your Google Play Games account (if you linked them).
I recently re-installed the game after a One-year stop, and it kept my progress (I didn't buy any features).