OllieJr,
Don't be embarrassed, the penis comes in all shapes and sizes. And, the size you have is not really all that uncommon. And, as long as it works for urination and sexual intercourse, that is about all that is needed.
Some men have a small penis when it is flacid, but it becomes much larger with an erection, as in your case. Some men have a large flacid penis, but it doesn't get any bigger with erection, it just stands up. So, again, as long as your wife and you are happy, that is all that counts.
As to the exam, you should not have to parade around in front of several people, naked. Usually, the physician, and maybe a nurse for assistance, is all that is in the room. If you happen to go to a teaching program, then you may have several exams by the different levels of physicians in training. But, usually when an uncomfortable procedure is involved, it is only done once. The other physicians are told what that physician found. On very rare occasions, the procedure may be repeated by the attending physician, just to make sure it was properly done by the resident.
But, if you see a physician in private practice, it is usually just the doctor.
And, the nurses and other ancillary staff are professionals and will act accordingly. If, at anytime, you feel embarrassed by what a nurse or tech does or says, say something to the physician or office manager. That kind of behavior is not tolerated.
If you prefer to be covered up as much as possible, ask the nurse for a towel or sheet to cover yourself. Your wishes can usually be accommodated.
Again, as long as you and your wife are happy with your size, that's all that matters. Try not to be embarrassed. The health care providers have seen all shapes and sizes. And, they should act like the professionals they are. In some urology practices, the nurses are male, as they commonly are in the OR.
Good luck. Hope everything turns out okay.