Be it that,were to, and should - what were the differences between these conditionals?
Solution 1:
This is just a tentative answer based mostly on my intuitions.
I think you have ordered the sentences according to decreasing likelihood that the condition is satisfied.
- gives no information either way, similar to modern in case; in that respect it's like the subjunctive in Spanish or French, which is why this use of be is called the subjunctive.
- suggests, and
- claims, that John won't ask.
Though the difference between should and were to is certainly not very marked. I may have got this part totally wrong.
(Note: The variant of 1 in which the main clause appears to contradict the condition has more or less survived: all be it [that] is now spelled albeit. Also, be it that was often used in the following construction: be it that A, be it that B. It means something like: *maybe A, or maybe B. Though I think it can also mean: maybe because A, or maybe because B.)
As today, by choosing will/shall or would/should in the main clause, you could select how strongly to suggest that John won't ask.
Interpreting the choice between will and shall in the main clause is enormously complicated due to a series of mutually contradictory prescriptive rules fighting with each other and with the natural use coloured by the original meanings of these originally lexical verbs. See the Wikipedia article. Therefore I have so far ignored the fact that you used shall (not should) in 1 and would (not will) in 3.
Whether will or shall is used for pure, uncoloured futurity probably depends strongly on the author and genre in addition to the period. If both 1 and 3 are taken directly from the same source, then there may well be a meaningful difference, though it's not clear what it is. It would be most straightforward if dutifully could be added to 1 and/or readily to 3 without changing the sense much, but that's not the only option. It really depends on the author's personal style. According to one rule that is sometimes still seen, the meanings of will and shall are actually reversed for the first person.