Recommendation letter submitted by applicant in application
Instead of asking the professor to give you the letter, tell him about your predicament. Mention that you checked with the internship program and were refused any alternative, and ask for advice. Be clear that you completely understand if the professor is not comfortable sharing the letter with you, and consequently would be unable to help you with this particular application.

If the professor is fine with you seeing the letter, he may offer it anyway. Even if he is not, it should not look bad on you, since you were left with no choice and were just trying to work around the program's requirement.