Just speculating on very little evidence because I don't use gotten at all, but for those who do use both
would "I've gotten" refer to the result of a process that takes time while "I've got" would be used where
the obtaining could be viewed as instantaneous?

I would use them thusly:

"I've gotten a bit heavy" - does imply it's taken some time, but also that I ain't done fattening up

"I got a bit heavy" - could imply a sudden heaviness (after falling in the pool and soaking all my clothes, I got heavier - awkward example, but one doesn't often get heavy suddenly), but could also suggest something like "While living in Italy and eating lots of pasta and drinking lots of Chianti, I got pretty heavy, but have since slimmed down" (Note: true story)