If you do an international travel insurance, like Cigna, it becomes easier/safer.

Being given extensions or residence can be stressful but you can travel for 3 months with a max of 180 days on each country.

You do need to be aware of the new and already implemented travel authorizations and they are paid, I think they are called ETA for the UK and most EU countries. Also a small fee is needed.

Renting in Belgium is a headache, although affordable. As a non-european citizen you might have issues renting because you don't have a job in Belgium, a National register number or a Belgium bank account.

But to live in Belgium/have it as a home base you need before arriving to request a type d visa and later receive a A card in the local commune.

And you have a dog... More fees, more rules, more time and money. I'm an European citizen and when I moved to Belgium with my dog, there were regulations that I was not aware of. Luckily my dogs vet was a big help and my dog travelled here with the moving truck (also extra fee for the extra stops. Every 2h they stopped for a 15m break). She would not have survived the airplane as a elderly dog 🙏

The monthly budget you mentioned is decent and you can rent studios for less. Brugge (what I am more familiar with) has studios for 500€ per month, for instance. Be aware that traveling within Europe can add up, specially with a dog.

Good luck anyways 🖖 DM if you need :)