Riley, I've never had any problem with RUN capacitors hold a charge - START capacitors are another story but most have a resistor across their terminals. The HF multimeter should be good enough to make your tests. While you're there, take a look at the RUN capacitor. It may be okay. It looks like your problem was due to a bad crimp on the wire from the Klixon but if there's any sign of deformity on the RUN capacitor metal can like a bulging top, it's no good - don't try to reuse it. It should look new, like it just came out of the box.
All you might need in the way of clean-up tools would be a small wire brush to shine up the terminals.
I mentioned prying off the old connectors with a screwdriver instead of grabbing them with pliers. You'll need something to pry against under the screwdriver and then slip the tip under the edge of the connector and slide it up and off. That way, they shouldn't be damaged. They should fit tight when you put the RED and WHITE wires back on the compressor as should the new connector on the new Klixon.