Nope. Tell your doctor if he ever has some time on his hands and gets really bored and curious, he should actually read the research on plant dermatitis, and the transfer mechanism involved. I have.
The allergic reaction is caused by the oils in the plant. Once the oil contacts your skin, it's absorbed in about 10 or 15 minutes. While the oil is still on your clothes or cat, it can be spread to new areas of skin, depending on how much is present. After that, your skin has absorbed all the oil it's going to, and nothing, not nothing, will spread the rash beyond the point of oil contact.
Period.
The reason that so many people, including most doctors, think that the rash can spread by itching or other kinds of contact, is that it takes several days for the reaction to progress to the rash, and it takes several more days for the reaction to complete. The first sign of rash will happen where the most oil contact happened, and the rash will progress over several days, sometimes as long as a week before the last of it appears. Also, the blisters release plasma when they're scratched open, which looks like something that should be able to spread something - but it's just plasma. Once the plant oils are absorbed by the skin, they are not released again later, any more than you could put lotion on your hands and a day later rub it off onto someone else's hands.
As far as excluding your cat, there are several things you want to consider - is it confined to your property, or does it roam free? If it can go anywhere, you always have the chance of it bringing something home. You should walk all the shady places on your land looking for ivy, and eradicate it. I don't normally advocate using chemicals, but for poison ivy I make an exception and buy the most evil roundup I can get legally. After hosing down the ivy and a one-foot radius around it, I salt the earth. I'm not kidding. You may decided such extreme measures are not necessary, that's just my process.
Good luck!
yay!!
You speak the truth sir. But scratching can cause folliculitis or cellulitis which is very painful. So pat or slap. Don't scratch.