In particular, I still don't have a counter to the fine-tuning argument which is short, assumes no foreknowledge, and is entirely intellectually honest.

The "fine-tuning" argument falls into the script:

1. Here is a puzzle that scientists can't currently explain
2. God explains it
3. Therefore God exists

If you accept that script you lose the debate, because there will always be some odd fact that can't currently be explained. (And even if it can actually be explained, you won't have time to explain it within the limits of the debate and the audience's knowledge.)

The trap is that it is a very temping mistake to try and solve the puzzle yourself. It's highly unlikely that you will succeed, and your opponent will already know the flaws (and counter-arguments) to most of the existing solution attempts, so can throw those at you. Or if you support a fringe theory (which isn't generally considered in the solution space, but might work), the opponent can portray you as a marginal loon.

I suspect that the theist wins these debates because most opponents fall into that trap. They are smart enough that they think that they can resolve the puzzle in question, and so walk right into it. By debating domain experts, the theist positively invites them into the trap.

How I might respond. "I can't currently explain the values of physical constants, and as far as I'm aware no-one else can either. If you think you have an explanation, you can do the scientific community a great service. Just formulate your 'God' theory as a set of equations from which we can derive those values, including some values or degrees of precision that we don't currently know. Propose experiments by which we can test that theory. Submit to a leading physics journal, and get physicists to perform the experiments. When you've done that, you can claim evidence for your theory, and I will be more inclined to support it. You can't do it though, can you?"