Hello and welcome to JustAnswer!
Removing a battery cable with the engine running can quickly damage electrical components if the charging system is working so I would recommend against that.
You may have a bad alternator, voltage regulator or a wiring problem.
First check the output of the alternator right at the output terminal and make sure it's actually low. If you find it's actually charging at the alternator but it's not making it to the battery then you have a cable problem or burnt fusible link.
There are two wires at the alternator that are needed for it to work. In the two wire connector there is a red and a brown wire. The red wire should have battery voltage with the engine running and the brown comes from the voltage regulator. Make sure you have battery voltage on the red wire, and you can very briefly apply ground to the other wire to see if it starts to charge. If it doesn't charge then replace the alternator.
If it does charge and you are missing ground then you either have a wiring problem between the regulator and alternator, a bad regulator, or a regulator wiring problem. Make sure the regulator has battery voltage when running, it's properly grounded at it's mounting point. You can momentarily ground the brown wire to see if it charges and this will rule out or condemn the control wire from the regulator to the alternator.