The dangers of very low doses of radiation -- the sort your pooch might face sitting on your husband's lap -- loom large in the minds of personal injury and medical malpractice lawyers but those risks are hypothetical and based on pretty wobbly science. The warning about
incidental exposure is not so much intended to protect pets and small children as to protect the doctors' insurance company from predation by roving bands of lawyers.
That said, if your husband decides to go ahead with the brachytherapy and is worried about
irradiating the dog you could buy some sort of radiation shield for his lap. If you google for "brachytherapy lead underwear" you will probably find something designed for the purpose. I find them catastrophically unfashionable. You can also buy lead blankets (the sort your dentist uses for x-rays) although you want to verify that the vendor will sell to consumers. Some of them only sell to medical facilities.
There is also a company that sells
a lead-lined apron on Amazon that is designed for brachy patients who worry about
pets and small children.
Oh, and I would be remiss if I didn't mention that Gleason 6(3+3) is never particularly urgent. Your husband has lots of time to look at options and make up his mind.
Welcome to the forum.