How difficult is it to break in from a low-tier BB?

Everything said here so far is accurate.

Your academic pedigree will really carry you, you'd be surprised. 3.9 kids are never considered dumb, so seeing that halfway down your resume will immediately quell any rising notion that you're at the bank you're at because you couldn't remember the answers to tougher interview questions. This is especially true for target schools. It almost paints the scene for a softball question "So, why'd you pick X-bank?"

If you get that from someone, you're gold. It's now a sales job. You need to explain with clarity and conviction why you had an interest in that specialized niche and just how lucky you were to be able to interview for it, get selected out of the 100-person analyst class, and be staffed on the deals you were. Capital raises aren't bad deals. Every single company has to fund itself to stay alive. If you speak about them in an embarrassed way or omit mentioning them altogether, you've made a mistake. Being able to speak intelligently about the why behind that corporate doing the raise in the way it did, not the simple what of the raise, will set you apart.

I realize now that I've omitted the most important point that makes my prior paragraph make more sense: you need to run your own process. The more specialized the group you're in, the more specialized your exits can be.

Here's a link to another post I made about a week ago where I described a guy that's turned into a pretty solid acquaintance that I hope progresses towards a stronger friendship. He parlayed a role in a group really similar to yours into his own literal dream job.

In your shoes, I would be finding every warm 'in' I could for every single specialist fund that matches your experience. Trust me; they'll take the interviews that headhunters set up, but they're going to be the generic 3.7/3.8/3.9 kids from top product groups at GS/MS and the EBs. Even the ones who 'focused' aren't going to be as specialized as you.

Ban from your mind the notion that your experience is inferior, because a partner at an infrastructure fund isn't as worried about your quantitative knowledge from executing deals as he is whether you understand all the 'why' levers that go into a good deal. The reason people value transaction experience is because it often correlates fairly positively with at least a directional understanding of those levers. If you can prove you grasp the levers, whether you have the experience or not starts to become an afterthought (as long as you know enough of modeling to not be an embarrassment).

Good luck.