Lock Mitre Cutters work very well - once you have it set up!
It can take quite a while to get it set up spot on so plenty of test cuts on the same size wood will be required. Once you have it set keep and mark the last successful test as a template. That way next time it will be quick and easy get the right cutter height.
there is another that will work on less that 1/2", but I cant remember whose- have a look through the trend, freud, and any other common ranges. I spent some time looking for this, only to struggle for nice hinges that could e used on 3/8" timber. might have been called a baby mitre cutter, baby mitre lock or something along those lines.
A 45º chamfer bit is fine. Are you using a mitre fence (or square of mdf run against the fence) to keep the end of the workpiece square. Its a case of adjusting the cutter height/fence and making test cuts
To glue up (or check your fit) try sticking the outside of the pieces onto masking tape and 'rolling' them up into a box