>Re-encoding them and/or truncating them is as hard and error prone as just generating the randoms yourself.

Feed the whole thing into sha256 and take the last X bytes where X is the amount you need for your shortener. If you can show that the last X bytes of sha256 have a chance for collision due to input, there is a lot of money in it for you. One of the nice things about a cryptographic hash is that all of the output has to be completely unpredictable based on the input.