Can a weaker version of the Halting Problem be solved?
If you define $H$ to take a 50-or-fewer state Turing machine (TM) and an arbitrarily large input, then if you can build a universal Turing machine (UTM) in 50-states, then you can simulate $H$ by encoding its description as well as its input into the input of the 50-state UTM.

If $H$ merely needs to tell whether any 50-or-fewer state TM halts with an empty tape as input, then, since there are finitely many such TMs, it can simply have a table hard-coded into it. Producing this table may be difficult, but it "exists" in the usual mathematical sense.1 Given values of the Busy Beaver function, we can actually easily describe one potential implementation. Simply simulate the input TM up to the appropriate Busy Beaver number of steps, and if it hasn't halted by that point, then it never halts.

To give an idea of how difficult it is to produce the aforementioned table, just note that we can write a program that terminates only if ZFC set theory is inconsistent. Gödel's incompleteness theorem then implies that if ZFC is consistent, then it is not possible to prove that that program doesn't halt. The upshot of this is after a certain number of states (and not a very large number) even if you were outright given the (correct) TM to do what you describe, you could not prove that it was correct in ZFC. (This isn't particular to ZFC. ZFC is just the formal system that is usually considered the "foundation" of mathematics. Other formal systems will merely have different cut-offs.) In other words, even if you were given the answer, you could not convince yourself that it was correct.

1 Actually, it exists in a stronger sense than many mathematical objects in that we can actually enumerate all the Turing machines and at some point the correct one will come up. We just won't realize it when it happens, in general.