"I'll be having to go".?

When I stretch a phrase out with
"I am having a headache" instead of "I have a headache" I am putting it in a temporary form and this is usually referred to as using passive voice. It implies a kind of hesitance or attempt at euphemism.

It is not wrong to say "I'll be having to go" but it sounds like you mean "go pee" because the passive voice implies a euphemism. Try saying "I will need to go" which is still passive but less awkward. Better yet say "I need to go tomorrow before four." Or whatever the specifics are.

"I'll be having to go" is grammatically correct. It's future progressive tense. But to use simple future "I'll have to go" is more common and probably considered more correct is more circumstances. Since you didn't give us any context, it's impossible to say which tense you should use.

Yes. Alternatives, I must go or I"ll be going.