Post-MBA carried interest negotiation if you have special talents at the VP level

I am at a MF and have been focused on healthcare for undergrad + an additional relevant degree (e.g., MS Healthcare Administration, MD, etc. - don't want to dox myself but it's a strong degree). I also did healthcare banking, short healthcare growth equity stint (<1y - partly a replacement of my third year of banking), then healthcare MF.

I have a particular focus on provider side. Because of this, I have a very deep rolodex of heads of the biggest hospitals, know all the relevant bankers, have strong network of most major relevant investors that we would buy from, and know the corporates who would buy our platforms. In a VP role, I would already able to independently operate around the ecosystem and drive dealflow and liaise with my proprietary set of industry partners in order to both conduct diligence, (potentially) source operating talent, etc. While I haven't done this yet as I have only been an ASO, I am comfortable with how I would be able to progress. I am now going back to school at HBS this fall.

I have been in talks over the last year or so with a brand new $300-500mm HC shop focused on exactly what I focus on. They have tons of advisors but no real talent below the MD level, and only one of their MDs has actual investing experience (albeit at a name brand fund) - the others are from nonprofit and/or provider side. They also have not hired much over the last year so I see this as an area they have of great need.

I could go back to my MF after MBA and rise through the ranks on our healthcare team, or I could go down market. I need to be appropriately compensated for moving down market though so I wonder how I go about the conversation of letting them know that I would need compensation equal to the MF in terms of carry dollars. Am I out of my mind or is this something I could negotiate? If so, how does this conversation go?