Random super bump, I didn't make a mod but I did find a way to force the prompts using some light reverse engineering.
Open the current steam build of SaintsRowTheThird_DX11.exe with HxD or any hex editor and edit the code at 0x001BA95B from "C6 86 F5 09 00 00 00" to "90 90 90 90 90 90 90" and save the file.
(you can also find this bit of code by searching for it, if you have different versions of SRTT or want to apply it to the DX9 version)
This will disable the game reverting the prompt icons from gamepad to keyboard/mouse while you are in-game.
If you alt-tab or never use a controller to begin with, it will revert use keyboard/mouse prompts again. Kinda neat if you like playing in both ways, since it doesn't force you into Gamepad unless you actually start using one.
However, it is worth noting that I recommend setting some button to switch the right touchpad to send right stick inputs as well as without that you can't rotate characters or zoom in on the map (as the mouse cursor no longer shows up in gamepad ui mode.
This is kind of the most hacky workaround ever but if someone wants to turn that into a mod, by all means (idk how to do that myself)