"But ... they'd sound ridiculous, if I presented them just using my southern English accent, with no recognition of the original accent and dialect."

Don't be so sure. I sing plenty of Scots and Irish songs and I sing them in my own (London) accent. As yet, I've never got the impression anyone finds them ridiculous.

Depends, I suppose, how embedded the dialect words are: if there are dialect words in every single line and none of them have an easy parallel in English English, well, it's probably not worth singing that song.

""What have you been doing (or where have you been) since the last time we met" does not have the same impact as "Where hast tha bin since I saw thee?"

Depends on the delivery. You don't have to say "bin" in whatever accent that is referring to: the difference between "been" and "bin" is very slight! And I personally would rather sing "thou" than "tha" if I was going to sing it as archaic language. So there are plenty of options.

You can do all sorts of pronunciation cheats too. There's a line in a song I sing: "Better if I'd never seen ye" and I sing it as "you" but slightly colour the pronunciation to imply an "ee" in order to maintain a rhyming line-ending. It works because there's a little of that "ye" in my South London accented "you" anyway.

Don't forget that it's not impossible that the Scots/Irish/Whatever song you are worrying about might itself have been transposed from another dialect anyway...