Lobbyists look for the gravy train. Not for a one-off, unstably funded contract job.

Well, there are lobbyists who will take such work, but you're getting second tier work at best. Unless they believe in you, personally.

I've been peripherally involved in such a relationship at the state level, and it worked out. Even then, though, there was a stable group of professionals who were organized and whose organization was able to contract with the lobbyist over a period of years.