I don't believe so. But even if there were, unfortunately, these things basically work the other way around.
If you can encode a problem in n states, and you know BB(n), then, as you say, you could use that to solve the problem. Trouble is, how do you know BB(n)? In reality, the only way to determine BB(n) is to go and solve all such problems; there isn't any other easier method that you can apply that would then let you get answers for these problems as a consequence.
So, the value of BB(n) will always be a summation of "we did all the hard work of solving all the n-state problems", not something you do separately to get those answers out.
If you can encode a problem in n states, and you know BB(n), then, as you say, you could use that to solve the problem. Trouble is, how do you know BB(n)? In reality, the only way to determine BB(n) is to go and solve all such problems; there isn't any other easier method that you can apply that would then let you get answers for these problems as a consequence.
So, the value of BB(n) will always be a summation of "we did all the hard work of solving all the n-state problems", not something you do separately to get those answers out.