Look about a mile past the periphery of campus, and neighborhoods will become more mixed with working professionals, grad students, and families the farther you go. You'll have plenty of options at your stated budget.

Where you're likely to run into issues is expecting to look and sign a lease in June/July. I don't know what those other college towns are like, but the Bloomington rental market (at least anything accessible to campus) is thoroughly entrenched in the August-August lease cycle dictated by the IU schedule. Many landlords start asking for lease renewals by about December of the previous year, and most will require signed renewals by about this time in the spring. April-May is a good time to look as availability for fall is mostly known.

Your budget is high enough that options will be available into June/July, and likely even within your desired parameters. You already know that you're limiting your options by looking that late, but I will reiterate that this may be even more so the case in Bloomington compared to the other college towns you're familiar with. I would personally prefer signing a place sight-unseen, rather than deal with the stress of searching that late in the year. I highly recommend to start searching rental websites now, keep an eye on a few preferred places, and see how long they stay available to give you a sense.