Faulty starter solenoid killing car?

What is "2v at the fuse?" Fuse link up by the starter relay? You have likely found it, either the link is AFU internally, corrosion, etc, or a bad connection right there such as the eyelet terminal on the relay end of the fuse link

Also you must be careful evaluating and taking voltage readings UNLESS you have a load on the circuit. Voltage drop is caused by current flow, and if everything is off, there is no flow. Things could show fairly normal and still have a bad connection