In most vet exam rooms there are signs that say something to the tune of: "For your safety and that of your pet, please allow them to be restrained by our trained staff"
It is there for a reason.
I know you think you are the best person to hold and comfort your pet...and you are. At home. Not at the hospital.
When pets come to the vet, esp when it comes time to actually do things to them they don't like (like shots, rectal exams, etc..) they are not in the same mind-space they are every other minute of the day. They are scared, freaked, stressed and possibly aggressive. Animals often become complete opposites of their usual selves when at the vet. Fear and adrenaline can make the most loving dog into Cujo or the cat even the owner can't touch into a submissive blob.
If your pet freaks out and struggles or becomes aggressive...the staff is trained to handle it. You are not. If your pet suddenly lashes out...it will not know who or what it is biting at. It just wants to get away and make the bad scary thing stop happening. It has no idea why these strange people in this place that smells weird are sticking things in his butt, poking him with needles and hurting him. Most don't understand that we are trying to help. They only know that strangers are hurting them.
So yes, Pookie may very well bite out of pure fear. And if you are the closest thing to him...you are the one that will get bitten. And Pookie will then attempt to dive off the exam table and potentially hurt himself worse then whatever you brought him in for in the first place.
So please. For everyone's sake. Let us hold your baby for just these few minutes. We'll give him back, we promise.