First I'm sorry to hear that you have cancer. I wish you a full recovery and I understand how it can change your perspective on life.
I have no specific recommendations for you right now. We sometimes travel with the dogs but have never needed a house that big.
If you use homeaway.com you can filter it down to pet friendly, price, neighborhood etc. Don't rule out using VRBO it just takes a little more effort to find the pet friendly places.
I'll be doing a little research for a future big family trip. What is your preferred breakdown for bedrooms? If I come across anything in my searches I'll let you know.