Any good or even half way decent vet will not anesthesitize your pregnant dog without fully consulting with you first. The x-rays can be done while they are laying on their back and/or on their side. Most any dog that is bred (has the health, tempermant, etc.. needed to be bred) will be co-operative for this x-ray. Not to mention your vet should jave the knowledge to deal with such an ordeal.
If you or the vet thinks that your dog is not a suitable candidate for such a procedure without the use of anesthesia then unltrasounds are also available. Close to the same cost now-a-days and easier.

And most vets have "pillow molds" in all shapes & sizes to form around your dog so he or she can sit still in that position more comfortably. So little to no training is required. If you feel training is needed focus on the stays & laying still on their sides & backs. You can practice this for all positions if you want just in case they ever need any other treatment done (it's good for training & bonding too).

Congrats on the litter!!! Big time!!!! If I was in the position to take on another dog I would seriously consider one of your pups.