Is there a copular equivalent of bheith, and if not, how do you say things like "to want", "to be able to", "to like", or even just "to be" (apart from bheith ina, if possible)?

"Is" has no verbal noun.
to want = bheith ag iarraidh
to be able to = bheith ābalta...
to like = dūil a bheith agat (in...)
to be = bheith / bheith ina

Actually I don't think you ever need to use a "verbal noun" of "is", there's always a way to say it otherwise. If it had been mandatory to express oneself, Irish speakers would have invented one.
The translation depends on the sentence.

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

Alright, so there are no exact nominal equivalents of is féidir, is mian, is gá, etc.?

Nope.
Do you have sentences to translate, where you'd need those? We would see, but I think in all cases there's a way to express the same thing without needing a verbal noun of "is".

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

No, I was just curious, but let’s say, “I’d like to be able to speak Irish.” Ba mhian liom bheith ábalta labhairt Gaeilge seems to work just fine, is there a better way to say it?

Should be "ba mhaith liom bheith ábalta Gaeilge a labhairt", or in some places (Cork) "ba mhaith liom bheith ábalta ar an nGaelainn do labhairt".

It can't be "ábalta labhairt Gaeilge".

You can also say "ba mhaith liom Gaeilge a bheith agam" (I'd like to have/know Irish).

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

Alright, thanks.

Féidir, mian, gá are nouns. So, in case you need a "nominal" form you can just use these nouns
(exc. for féidir, of course, which has lost all its meaning outside of "is féidir/b’fhéidir") but not as a kind of infinitive. ("to be able", "to want" etc.)
Even in English there's no "to can", "to must" etc.

Yeah, alright, thanks.