That explains the weirdness that's not really a mobile hotspot. That's supposed to be for fixed wireless internet. The battery is just there to act as a UPS.
Just be careful to not drain your car battery. Lead acid batteries are not fans of being used that way.
If the inverter doesn't work out this should work to power the wireless internet.
https://www.walmart.com/ip/CyberPower-CPUDC1U2000-power-adapter-car/30253317
This will power the laptop