I would follow your doctor's advice and get the biopsy ASAP. Something is causing an elevated PSA, and only a biopsy can rule cancer in or out. Has he tried a course of antibiotics to rule out an infection?
I'm sure you understand that (1) an elevated PSA doesn't mean you have cancer, but it shows something is going on in the prostate. It could be an infection. The point is, you need to find out what's up, and (2) the fact that you feel healthy doesn't rule out prostate cancer. Many of us felt perfect before learning we had prostate cancer.
If your wife found a lump in her breast, I'm sure you would tell her to have it checked out right away. I would give you the same advice. Get it checked out.
Have you had a digital rectal exam? Sometimes the urologist will feel a lump or something unusual that might further indicate a problem.
In any case, it's always better to know and have the problem taken care of, than to pretend it doesn't exist and hope that it doesn't mean anything. Nobody WANTS to do a biopsy, but it can be a lifesaver, and in personal experience wasn't a painful procedure.