Yes, some of the lower DII recruits would appear on your DIII screen if you had better prestige.
Also, if those players are unwanted at DII, some of them will drop to the DIII list. They drop to the A+ prestige teams first, then A, A-, etc.
Place a phone call to any DII players you're interested in. If they'll possibly drop to your school, you'll get a 'backup' message. If they insult you, or don't respond at all, you have no chance.