M7 Odds, Wharton possible? LMM PE, limited pedigree

I made the jump to PE a few months ago from IB. No name LMM IB to LMM PE ($300M fund, large city). Enjoying it so far but unlikely to be direct promote to VP since everyone VP and above has an MBA – all from M7 schools. Some HSW but mostly from the other schools.

This question might be pointless without my GMAT score, have not taken it yet.  I got a 1400 on the 1600 SAT scale for reference, so I would likely need to study a good bit to get a 750+ - willing to put that work in…no guarantee of course.  I would like some feedback on my candidacy since I will probably start to study for the GMAT soon since ~2 years is gonna fly by.

• White male, middle class upbringing.
• 3.57 GPA, large state school. Graduated with honors (barely). Transcript is a bit sloppy…some semesters I had a 4.0, others dipped to ~3.3
• First job was not great, made jump to boutique IB shortly after. Spent 3 years there, first direct promote to associate there but was barely associate before leaving IB.
• Thrilled to be at a PE firm with a fund. Not to blow smoke up my own rear but headhunters told me “Candidly, I’m shocked you are getting as far in PE processes as you are with you bank brand name, no offense. You should be proud” (paraphrased).
• Limited volunteer work but getting more involved now (found it tough with IB hours/variability…). I volunteered a lot in HS and college so hopefully a recent ramp doesn’t look fake, the group I volunteer with now I am fairly passionate about so I don’t think it will come off as hollow.  
• Will be over 30 by time of enrollment...is this a major ding?

So a white guy with finance experience, limited pedigree. Some hair on my candidacy (college transcript, meh first job), but some people eat up the scrappiness…clawed my way into PE.

I think H/S are out of reach/maybe not worth the apps but W might be possible if I kill the GMAT (I tell myself). Unsure about the other M7/top 10.

Any thoughts on my preliminary business school profile? I’m a few years away but appreciate any feedback.   

Edit: Willing to pay for a quality admissions consultant too, when the time is right. Relatively small $ for potential large upside.