If I want to use "bountying" in a sentence, how might I construct a spelling for it? If not possible, what word could be used instead?

First of all, you could always say that you post (or award) a lot of bounties. This seems the most natural way to express it.

But if you’re still looking to make bounty into a verb via zero-conversion, it might interest you to know that it doesn’t have a common phonetic pattern for ending verbs with. Indeed, English has no verbs at all that rhyme with bounty, and it has only two that end in ‑nty at all, neither of which is in common use.

English doesn’t even have all that very many that end in ‑ty either, but some of those in common use are pity, empty, and dirty, and rather less formally the twencen American colloquial verb party meaning any or all of attending a party, having a good time, or partaking in recreational drugs, is probably well known by now.

In any event, there is no question of how the putative verb’s -ing inflection should be spelled, as these are all perfectly regular, even crying, plying, pitying, partying, buoying, conveying, reifying, quaying. It’s not like money and monied.

That leaves you with the same bountying you started with, if that’s your druther. All ‑ing verbs are perfectly regular, at least phonetically. Occasionally orthographic tweaks are needed, as in betting, braking, dying, eying, picnicking.