There are several ways to go about doing that.
1 - Leave the sticks in there. You're making an arcade stick, so you have piles of room, but if you must make more work for yourself that's your call.
2 - You can also leave just the POTs from the Sticks on there and turn then to the center value and lock them in place with some hot glue. If they are already removed or damaged in some way, then...
3 - Use fixed Resistors, anything from 1k to 100k will work, as long as they are all the same value. You need 2 per POT, so 8 total for doing both sticks. Put one in each outside hole where the original POT was and then both go to the center hole.
O--10k--O--10k--O
If the controller was calibrated with a real janky POT or it drifts even with Resistors in there, then..
4 - Use 4 TrimPOTs (Trimmer Potentiometers) vlaue 10k to 100k and the more turns it has the more accurate it will be, in place of the stick POTs. These cost more than Resistors, but they also let you 'dial in' the center value to keep it from drifting.