Brief:
Generally the words "budget" and "Tokyo" do not belong in the same sentence. Tokyo is a VERY expensive city. Now, about gay venues: you'll be hard pressed to find any...and if you DO find them, to be welcomed inside unless you're with a Japanese guy. The Japanese often view anyone other than their kind as unwelcomed.
However, if you CAN find a Japanese guide, he'll doubtlessly be able to direct you to places where westerners are welcomed.
Good luck and have a good trip.
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