Should I request a biopsy for an itching and tingling mole?

It's likely your dermatologist felt that your skin lesion had very few characteristics warranting a biopsy, but if there are any concerns that it could be cancerous, a biopsy should be done.  Itching and tingling does not automatically mean a skin growth is cancerous.  It sounds like a biopsy will at least set your mind at ease, so ask for one.  The procedure should be done by a competent, licensed provider with adequate knowledge and experience to ensure it is done properly and with minimal scarring.  I hope this helps.
In health,
Dr. Derico