It's your choice.
Absolutely, it is your choice. First and foremost, you need to do what makes you the most comfortable. That being said, you also need to work within the bounds of what your particular doctor's office will permit, or what they will find acceptable.
My preference would be to strip completely, including removal of all underwear and socks -- completely means everything comes off. Then either cover up by laying the gown or a paper sheet over your lap, if either is offered. The nurse will likely give you instructions as they are leaving, telling you how much clothing to remove (or not) and handing you a gown or a gown and a sheet. Based on those instructions, and your level or comfort at that moment, you can make your decision. You can choose to completely ignore the nurse's instructions and just do what you like, or you can address the instructions. By that I mean, if they tell you to leave your underwear on and put on the gown, you can nod in agreement when they're in the room, not causing a scene or arguing with the nurse, but then strip completely after they leave. Or, as the nurse hands you the gown, you can engage them in conversation, telling them that you probably won't use it since you feel more comfortable without it. Just say that you've found that you're much more apprehensive about the exam when you have to wear a gown and that you're more relaxed if you just get all of the undressing out of the way from the start. Whenever I've said that they usually acknowledge my preference and indicate that it's fine, that whatever makes me feel better is OK with them. Of course, once the doctor enters the exam room, there's really no need for a covering at all. If you're really brave, you can set the lap covering aside completely, saying something like "oh, I guess now that the door is close we won't be needing this at all".
Good luck and please let us know how it went. 😃