Honestly everything checks out except for the international PE part.
I think you're targeting the right range of schools that you'd be competitive for. GMAT is fine - of course if you feel you can boost it by 20+ points without much effort, it may be worth it to give it another go in the next few months since you have time (and still be able to make the R1 deadlines in Oct). Shoot for say a June/July exam date. But again it's not essential in your case since your GMAT is good enough.
As for the international PE - I wouldn't talk about that. Keep in mind that with PE, they are almost exclusively looking for folks with transactional/deal experience, and those tend to be investment bankers and PE associates prior to b-school (and there's more than enough of those folks at b-school that funds don't have to look beyond this very specific segment of folks). If PE is really what you want to do, you are better off working in banking -- even at places like Carlyle where they have more foreign affairs folks at the senior levels, they want junior folks to be the ones doing the grunt work (modeling, due diligence, analysis, term sheets, etc) and with ex-bankers and ex-PE guys, it's plug and play and complements the senior partners' skillsets (network and foreign affairs experience).
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