If you find yourself having to look rather far outside the center and/or increase your budget, you might consider checking Premier Inn availability. You'd need to go to the corporate website rather than using booking.com (though rooms occasionally are offered there). The non-central Premier Inns may have quite reasonable rates, and many of those hotels (not 100%) are air-conditioned.
I just randomly checked rates for 4 nights beginning on July 1, and there are seven PIs under £100 per night (on average). I didn't check to see which of those has air conditioning. The low rates will be non-refundable; the flexible rates for those hotels go up to about £110 per night. £110 is only $146, which is within your target range. Rates can vary greatly by date, of course.
Unfortunately, PI runs good hotels but a rather bad website. The dates you enter will keep changing as you click from hotel to hotel. If you decide to book something, I recommend using the smartphone/tablet app, which I think behaves well. If you must use a desktop computer, keep watching the dates and changing them back to what you want.