It’s sometimes hard to avoid the “louder sounds better” pitfall when it comes about judging your (live) sound before and after you used ableton to polish it.
Try to listen to the dawless mix and your ableton mastered version at the same levels. Does it still sound better and if so, why?
Then you can adjust the sound you like with your dawless gear only, by using the tools your setup has. Process samples if needed (or resample them).
Note that most of the time in a live situation the requirements of how your sound is mixed is different as for a mastered recording. Think of stereo imaging, frequency range, the amount of details, etc. You can use this in your advantage.
There is no need to “max out” the levels of your music when you are performing live with your gear. That’s also the charm of live: That you can have a different sound and have more dynamics compared to fully polished squashed tracks.
Do you play after a DJ who maxed out all gain and just played a -4dB RMS track? Start with an interesting intro / sounscape first. Ears calibrate volume within a minute and it makes your liveset more interesting too.
Then in the end you maybe don’t need a “mastering chain” after all.