Top Law Schools

I don't want to out myself so I can only describe high-level details. I'm a relatively recent graduate (3-6 years out, T35) working on the policy/business side in Big Fed (FRB, FDIC, OCC, SEC, CFTC) in DC. I don't have big law experience but am interested in transitioning to an FS regulatory practice group. Before Big Fed, I worked in a multi-client FS-policy-oriented role (further description may out me), so I haven't practiced law. Love DC and would prefer to stay long-term, but open to other markets if better opportunities.

My work is relatively niche and deals with specific Dodd-Frank provisions (further description would out me), and my group's portfolio is large institutions. I'm not in legal, but work closely with them often (rulemakings, guidance, testimony, speeches, reg/legislative analysis, etc.).

I'm aware of the pros and cons of Big law vs. Big Fed, and know that I want to explore my options with big law. My questions are:
1) Recognizing my lack of big law experience (or even legal practice) puts me at a disadvantage, does policy/business side transfer to an FS reg group, or would I need to work in legal at the/an agency? I'm willing to class down, if that matters.
2) The current environment will obviously limit hiring/networking options for the foreseeable future, but should I reach out to specific firms? Are some firms more open to non-legal practicing backgrounds then others?
3) Excluding obvious ties (friends, alumni, etc.) should I reach out to practice group heads, partners, or associates?

Feel free to pm me if you need/want more detail, thanks in advance for the help.

Not sure about how a policy gig translates, but I will note that those listed agencies have pretty different placement power to biglaw, with SEC clear in the front (and CFPB, which you didn't list but fits the description of your role, in the back). FDIC and OCC regularly hire from biglaw and have people place from there into biglaw; no real intel on FRB or CFTC. I would consider reaching out to a DC recruiter to pick their brain about how best to sell your particular experience and which compromises (class year, practice area, geographical preferences) are worth making. I'd also see about chatting with anyone currently in biglaw who started in FS policymaking (or did a stint there between other gigs).

Thanks that's helpful, I'll look into recruiters in the area. Anyone know which firms have had folks from policy make the transition/which firms tend to hire more from agencies for their FS reg groups?