Given the choice between ending the original sentence with "better better" or "better, better," I would probably go with the commaless form:
I have spent my entire life making that which needs to be better better.
At the same time, there is nothing incorrect (in my opinion) with marking an intended slightly-longer-than-usual pause between the two occurrences of better with a comma.
Having said that, I must commend the comments of Edwin Ashworth and Janus Bahs Jacquet (above) that you could easily avoid the whole question by rewording your sentence as (per Edwin)
I have spent my entire life making better that which needs to be better.
or (per Janus)
I have spent my entire life improving that which needs to be better.
To the extent that your decision to present the reader with a double helping of better reflects a desire to call attention to their juxtaposition—as opposed to, say, a failure to notice that less flashy or distracting alternatives are readily available—I imagine that you don't mind causing the reader to stop momentarily to take in the duplication. And since this entails forcing a pause in the reader's progress at "better better" regardless of how you punctuate it, I don't think the comma serves much practical purpose here.