Is it correct to say: "I would do something, be it me"?

No. It sounds old-fashioned, but searching through Google books for such phrases used in the 1800s, I did not find a single instance of anybody using "be it me" or "be it I" in this way. What you're looking for is "were it me" or "were it I"; this usage requires the past subjunctive and not the present subjunctive. Here are some examples:

Why, were it me, I'd take it just to spite him.

Were it me, I'd show him the difference.

Were it me, I do believe I'd be half inclined to stay, and be discovered.

"Were it I," said the second man, with a shrug of his shoulders, "I'd get it to the Admiral,"

I can make nothing of it, Roland, but were it I, I should ask my mother exactly what it means.

Were it I that sold that poor perished victim his drams, I should seem to hear his voice mingling its wailings with the storms of every bleak wintry night.

Some grammarian prescriptivists would say that it should be "were it I", but both forms seem to have been used. This is still an old-fashioned way of saying this, and I recommend saying "if it were I" or (maybe less grammatical, but more common) "if it was me" unless you want to sound like you're speaking archaic English.

You could say: "I would do something, were it I"; you might even get away with the colloquial but incorrect "were it me".

But this, though correct, sounds like quite antiquated English, especially in speech.

Slightly more natural would be to say "...something, if it were I", but again, the incorrect but colloquial "if it were me" would sound even more natural.

Trouble is, even to use that last construction indicates knowledge of the subjunctive; everyone else you speak to will probably try and correct you to say "was", and if you contradict them you'll probably be marked for life as a pedant, and not welcome in anti-intellectual society.

In such cases, you use the past subjunctive, which is used for imaginary or hypothetical conditions. The present subjunctive is used to express commands or requests.

If I were rich I would live on Long Island.

This sentence express a hypothetical condition, as I am not rich.

I demand she be on time.

This sentence express an order. The verbs that express an order, and that generally use the subjunctive are ask, demand, insist, recommend, request, and require.

To notice that the past subjunctive is not called so because it is used for an event happened in the past, nor is the present subjunctive called so because it is used for an event that happens in the present.