Best Route to PE - Stay at MM with good team relationships, or lateral to BB?

I'm currently an analyst at a low-to-mid tier MM (think SocGen, BNP Paribas, etc), hoping to make my way to PE. Buy-side recruiting at my firm is pretty much nonexistent, and very few, if any juniors receive buy-side offers. I'm in a smaller group with consistent deal flow, so the analysts do get transaction experience, but the sample size for previous exits is too small to draw any conclusions. However, the MDs have been very pleased with my work, and I've managed to develop great relationships with all the seniors in my group. While I haven't brought up recruiting to anyone, a few of my MDs have buy-side relationships from earlier in their careers, and have even mentioned that they would support me recruiting if I were to show interest down the line.

Is the support I could get from seniors at my current bank going to be enough to overcome the bank's reputation during recruiting, or am I better off attempting to lateral to a BB/EB? I understand that my resume will be stronger if I lateral, but I also don't want to burn bridges I've built up so far, and even risk getting blacklisted from PE firms, where my old MDs have connections. I've gotten mixed responses from friends recruiting for / currently in PE, some of whom said that a senior's recommendation is more important than the bank you came from, while others said that having a well-known firm on your resume is going to be much more helpful.