Tuning should allow NUTS to find a reasonable step size near the posterior. Are you seeing failures during tuning? Setting unstable values to -inf might work, but if your likelihood fails with large values it seems the prior is at fault here.
You can also try to tweak the initial point, but I don’t think that would suffice for your case.
Not important, but the smallest step size with a proposal of 1 is actually smaller because it can go into negative domain now, so it would be e-700 - e-701 (exponentiation underflows around ~ e-712 in float64 so you can’t go much more negative).