You used to be able to use could, would, and did in constructions like this. And in fact, the sentences I found below don't sound ungrammatical to me, although they do sound quite old-fashioned. Your proposed examples are all ungrammatical, because they don't belong in the subjunctive mood.
Searching 19th century texts, we find:
Could I find aught worth transmitting from so remote a region as this to which I have wandered, I should gladly send it;
Could I see him, I might on this side guard against his penetrating eye, or on the other side act something in secret, safe from his inspection.
Nothing is more common among men ... than to say, "could I but live to see such and such an event take place, I should die content."
Could I but find a substitute for matches, I might strike a light (he had left the candles with me in very derision), and then escape was certain.
You can also find did used this way, but it seems rarer than could.
nay, torture itself could not be misery to me, did I but know that she was happy.
Surely no sinner would willingly incur eternal damnation, did he but really consider what it is
This is a past tense subjunctive mood construction, which rules out all of your suggested examples, as the condition must be counterfactual (or at least be likely to be counterfactual).
You can't use have, because it's present tense.
The remaining past tense auxiliary verbs are might and would.
I believe the fact that this must be counterfactual means that might doesn't work, as it's hard to see how a counterfactual would work with might.
This type of construction is also hard to form with would, as in this case, would must mean if he were willing to. However, you can find it:
A palace to dwell in—coffers swelling with gold—honors unnumbered—office, dignity and favor—all that could gratify ambition, taste or desire, were within his grasp; would he but accomplish one end! And what was that? To plunge another in misery ...
He moved a few steps nearer, and those few steps were enough to prove in how gentlemanlike a manner, with what natural grace, he must have danced, would he but take the trouble.
Why has the list of grammatical inverted conditionals winnowed down to had, should, were today? The slow death of the English subjunctive means that these are now essentially frozen forms which are not perceived as subjunctive, but which are just idioms. So it's not surprising that the rarest modal verbs in this construction (definitely did and would; maybe could) are no longer used today.