The only controversy that I recall over the word 'Moor' is when breweries have renamed pubs called 'The Blackamoor' or 'The Moor's Head' because they thought - reasonably IMO - that it might deter potential customers. The predicable 'outrage' (© The Daily Mail') does not come from folk who find the word offensive.

One reason for changing that first line might be to make it more understandable to a younger audience who don't know what a Moor is or, if they do, misread it as a racial insult.
"Girls came unto the door I look as black as any Moor"
It implies he's attractive to girls despite being a dirty sweep. So translate that for your audience.
"Though my clothes and face be black, it's not for girls I lack."
That's not great, but you get the idea.