If you go here, you'll see that, with beag, for example, the genitive masculine singular is big, but would that ever actually be used just like that (i.e. unlenited)? I know it's lenited after the noun and it can only come before the noun in poetry, so can it ever be used by itself, or are there specific cases where it would be used?

I don't think so. They've given it with no lenition there because it's out of context, it's not after a noun, so grammatically there's no reason to lenite it. But of course, in any real context other than quotation in a grammatical book or in a dictionary, it would be after a noun in the genitive singular, so it would be lenited in all cases.

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

Maybe it is possible to avoid lenition by separating it from the noun with an adverb?

ainm an fhir go réasúnta big

not sure if it would work this way, but it came to mind as a possibility

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ЯГОН ТОҶИК НЕСТ ИНҶО???

No, usually no genitive form following adverbs:
ainm an fhir réasúnta beag

For some reason, having beag before réasúnta sounds more natural to me, is this in any way an issue?

ainm an fhir bhig réasúnta

This would mean:
the name of the little reasonable man.

But "the name of the fairly little man"
ainm an fhir réasúnta beag

Wouldn't this mean "the name of the fairly short man"?
My impression was that beag when describing people meant short, not young

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ЯГОН ТОҶИК НЕСТ ИНҶО???

Yes, "short" in this context. Funny English language

GRMagaibh, was mnisinterpreting "réasúnta beag" as reasonable + small instead of "reasonably/fairly small".