Advice needed on PE offer

Looking for some honest input on a crossroads I'm at.

Background: Graduated from a semi-target in the UK, and started in a MO role at a BB (GS/MS/JPM) in London and obviously regretted it so managed to move to MBB and relocated to another European country (not a major financial hub). Now I have the chance to join a local PE fund as a PE associate.

The fund isn't big on AUM and isn't a household name outside the local market, but the team is genuinely top talent and the fund is well regarded within the country. It's the kind of platform that's respected locally but wouldn't ring a bell to anyone screening resumes in London or New York.

The dilemma:
• Take the PE offer now and get real buyside experience, which is what I've been aiming for
• Stay at MBB, build the brand and optionality, and potentially do an MBA to reset into a larger/ more recognized fund afterward

My questions for you:
• Is jumping to a smaller, non-name-brand local fund worth it over staying at MBB, given the buyside experience is what I actually want
• Does joining a fund that isn't well known hurt me down the line, especially for MBA admissions, since adcoms and recruiters lean on brand recognition
• If I do the fund now, is the MBA still a realistic reset button later, or do I close doors by stepping off the MBB track

For context on goals: I want to end up as a PE investor long term. The local market is where I'm currently based and the fund is a strong fit for it, but I'm weighing local relevance against international brand.

Appreciate any perspective, especially from people who've made a similar non-hub or non-brand-name jump.