Institutions consult roommates because allergies and genuine phobias are real and reasonably common. That consultation is not a vote on whether you may have your animal, but it can determine who you live with, which is a decision worth influencing early rather than discovering in September.
The students who find this easiest are the ones who raise it themselves before the institution does. A short message to a prospective roommate explaining that you will have a support animal, and asking whether that causes them any difficulty, resolves the question in a day and on friendly terms.
Where a conflict does exist, the usual outcome is a room change for one of you rather than a refusal. Raising it in spring means that change happens on paper. Raising it in October means somebody moves in the middle of term.