I only know how to re-map the button through Steam, but not in LEGO Racers directly, so this might not help at all, but here you go:
1. Start up Steam Big-Picture mode
2. Click on the gear at the top-right of the screen
3. Go into Controller Settings and make sure you've enabled "[x] Configuration Support" ([x] being the type of controller you are using).
4. Make sure your controller is plugged in. I recommend doing this with your controller plugged into your PC directly, just to be safe (it might be possible to configure your controller through Steam Link but, as somebody who doesn't own a Steam Link, I've never tried it before)
5. Exit Controller Settings and go to Base Configurations
6. Select "Desktop Configuration"
7. Change whatever you want the Start button to be binded to
8. With the required change/changes applied, close Big Picture mode (but don't close Steam)
[ NOTE: according to Steam, this change will be reset when you close/open Steam. Exporting and loading the controller configuration might fix this, but I haven't tried it. ]
Hope this (somewhat) helps.