Thank you so much everyone for your help but I am totally confused, I know the saying
'Over the rainbow' is probably not the easiest to translate, could you tell me exactly what I would need to write down for the tattooist? I really dont mind what type of Irish it is, I am just very grateful for your help.

I think we've agreed on

Thar an mbogha báistí.

I don't think any Irish speaker, regardless of dialect, would disagree with that as a good translation.

That is fantastic, thank you so much everyone!

Báisteach isn't considered adjectival here; its a separate noun.

In archaic Munster Irish (especially if the adjective begins with the same consonant as the noun) you do find adjectives nasalised/ ecliped following a noun in the genitive plural, or dative (really accusative) singular + def. article:

e.g.
ar an mbord mbeag
Tigh na mban mbeag

There's a few instances of these forms in the writings of the Blasket islanders.

The same is true for Old and Middle Irish.

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Please wait for corrections/ more input from other forum members before acting on advice

Thanks for your explanation. It makes sense for what I've seen but is, of course, archaic usage.

Báistí is a separate noun, but to some degree such indefinite genitives are treated as if they were adjectives or better: parts of a compound. Bogha báistí is such a compound (compare English rainbow written as a single word).
Such genitives usually share the same rules of lenition with real adjectives.
So, if adjectives would be eclipsed (at least in the older language) the question whether indefinite genitives could be eclipsed, too, is standing to reason.

But I'd think the answer is no.