If you love your companion animal that much look for private landlords that will allow you keep a pet, they're not as rare as people like to make out. Also you could get a pet sitter who will look after your animal for you whilst you're away at university and allow you to take them back during the holidays and after uni, if you privately advertise on gumtree or something similar I can assure you that you'll find someone. My friend Maxine and her family looks after peoples rabbits, guinea pigs and cats if they're going away to uni or if they're leaving the country for 2 months or more with work/business and offers this service for just the cost of the food and any vet bills that are required. There are kind and compassionate animal people out there, you just need to seek them out.
The final alternative is to consider permanent resettlement although I understand this would be difficult and you probably already know the route of this and how to go about it.
I would not recommend taking anything bigger than a dwarf rabbit/guinea pig to university halls.