Assuming you can accept measurements taken along a straight line:
Mount a digital dial indicator, with a serial/USB output, on a carriage that runs parallel to the blade.
You'll probably want to drive the carriage with a lead screw, to simplify anti-backlash. It's a short travel, and a light, so it won't be terribly expensive.
It's going to be extremely important to hold the blade precisely parallel to the track. I'm thinking two V-blocks under the edge, and a single spring-loaded arm with a V-notch at the top. That'll need to be designed with zero slop.
The V-block material will need to be selected for a trade-off between not nicking the blades and the blocks getting worn out of precision. I'm thinking Delrin, but I'm an amateur, so don't give my suggestion a whole lot of weight.