I do this all the time, and find that very often the counter staff pay little attention to my airline info (usually just my self-printed reservation confirmation, since most flights are now ticketless). You might even be able to get away with casually presenting an old ticket confirmation, or reserving a refundable r/t ticket, printing out the confirmation, then cancelling it.
Otherwise, most of the major rental companies have specific policies concerning the use of debit cards. Often the policies are not uniform across the company; some counters may accept debit cards, others may not. It can get a little confusing and you may have to do some digging. Look for the "frequently asked questions" or "FAQ" on the company websites.