I first used SDL2 Gamepad Tool to generate a valid config (other programs can do this too, antimicrox for instance), then I added a tilde (~) at the end of my right analog's X and Y axes which inverts the inputs. Last, inside Lutris I added the environment variable SDL_GAMECONTROLLERCONFIG for the game, and pasted the long config line as the value. I'm sure all game launchers support custom environment variables.
For reference my controller config looks like this (all one line):
030081b85e0400008e020000140100001118654Controller,Xbox 360 Controller,platform:Linux,a:b1,b:b0,x:b3,y:b2,back:b6,start:b7,guide:b8,leftshoulder:b4,rightshoulder:b5,leftstick:b9,rightstick:b10,leftx:a0,lefty:a1,rightx:a3~,righty:a4~,lefttrigger:a2,righttrigger:a5,dpup:h0.1,dpleft:h0.8,dpdown:h0.4,dpright:h0.2,