I'm another who has gone through several routes with different cats, including transdermal. The radioactive iodine treatment worked very well on Nougat, who was perfectly awful to pill, and she lived another three happy years before dying of rapid onset renal failure. The Tapezol pills have worked very well on three other cats whose age and/or other medical issues made them a poor subject for the iodine treatment. Alexander has been on the meds for 18 months now, and my Ginger was on them for 5 years (she was 17 when diagnosed and so considered too old for the radio-iodine treatment). They have/had no problem at all with digestive upsets, diarrhoea, etc. I think the transdermal does work, but it is not as effective, at least that was my experience.

I must admit that in your place I wouldn't want to risk the gift you've both been given of extra time for Sydney by having her go through the radio-iodine treatment. It is stressful for the cat, no doubt about that. And, unless the protocol has been changed, even when she is back home you won't be able to spend more than an hour a day with her for a week.