I can speak to it somewhat because I do it in humans for hyperthyroidism and thyroid cancer and we had a feline go through it. We had a 12 y/o female cat that developed hyperthyroidism, we tried the oral suppression for a year, she had terrible GI issues with it. I pushed our vet hard to refer us for i131 and he was reluctant, but there is a center in Irving that does it. Standard precautions like with humans, the half life of I131 is about 1 week, but the effective half life including excretion is about 3 days, they are only going to keep the cat for a few days after administration - they will be radioactive when you get them back. I can't recall how many mCi they gave but it was minuscule compared to human doses. Keep in mind iodine (and the body does know the difference between I131/radioactive vs normal Iodine) comes out of every pore of the body - feces, urine, saliva, sweat. Just keep them separated in a small room - we used an extra bathroom, covered the floor with disposable pee pads, disposable litter boxes, disposable water and food plates. Use gloves when handling and bag all the waste - all straight to trash. Ours spent a couple weeks in there - keep them away from the kids - the longer you can keep them isolated the better - 5 to 7 half lives and it's basically gone so ~ 15 days if you can keep them isolated that long is great, 3 weeks is fantastic.
All that being said it was a fantastic result for her - she lived to 17 years (another 5) with no need for any thyroid meds.
I deal with radioactive humans every day, just be smart, use disposable gloves, line the room floor and toss everything they use during their isolation. The risk to you or family is minimal but just be smart.