When I did my LLM, I had to leave my dog with my sister (who took good care and made my cute little dog fatter hahahah). But, yes, no college accepts pets if they are not service pets. Therefore, I would suggest you to rent outside of college.
Saying that, you for sure will find rent below 2k outside Uni properties. There are some surroundings with private owners and, right near the Cambridge Train Station, there are new apartment buildings for private rent. So I believe you will find some place, either paying more and closer to the city centre or paying less and some minutes more biking.