The charging compartment cannot be charged when the data cable is plug

It sounds like either a bad battery or a faulty lightning port (The original airpods use those ports, right?). Unfortunately, there's no way to open up the case to fix this sort of thing without mangling it to shreds. See this teardown for details: AirPods 拆解

If you still want to try, go ahead. If you happen to have some magic way to crack the case open without actually cracking it, well, it sounds like you don't really have anything to lose. Otherwise, I would try a different charger, and try the one you're using with another device just to see if it's actually the cord acting up.