Drum Switch
First, you need to do a little recon. The proper wiring should have the drum switch connected to the low-volt "logic" connections on VFD. There should be no high voltage leads whatsoever connected to the drum switch. If the drum switch is connected to high-volt power wiring, then it should probably be re-wired.
As you say, the VFD should do all of the high-volt starting and stopping. You should not be switching high volts on either the input or output side of the VFD while it is in operation. High volt wiring should go straight to the drive input. The drive output should go straight to the motor with no switches or disconnects. (At the end of the day, it's OK to remove power to the input of the VFD via a disconnect or a plug after the drive has stopped).
If the drum switch wiring is correct, everything may be normal and good. I can see that if you leave the drum switch on, and then use the keypad to start and stop, the drive will be getting conflicting inputs and will ignore all inputs until you reset it. If the drum switch is wired properly, you should use it (not the keypad) to start and stop the drive. (Its a lot easier to find the drum switch by feel than it is to search for the button on the keypad). Note: on most drives you can disable the keypad to allow exclusive control by an external switch - but it's easier to just remember not to use the keypad.
There could also be an improper logic setting in the start/stop logic, but I am thinking it is set correctly. Try the above first.
Over Current on Braking
Depending on what you find out about the drum switch might change my response to the OC (over current) issue, but let's assume that the drum switch is properly wired on the low-volt logic.
When you tell the VFD to stop, it can be programmed two ways: 1) coast to a stop, and 2) decelerate to a stop.
It sounds like yours is set to decelerate. You can program in how long this decelerate is: 0.1 sec to 999 sec, or similar.
If you apply the brake while the VFD is still trying to wind down the motor, the brake will generate high torque just when the drive has little to offer - and will likely trip the drive on "over current" since the drive will try to hold the motor speed at the programmed decel curve. I have the VFD on my mill set at "coast to a stop" to allow immediate use of the brake without tripping the drive. There is no real reason that I can think of to have a manual mill set for "decel to a stop." (If the mill were running without a VFD, it would always be coasting to a stop).
Look in the manual to find the listing of the various software parameters in the drive. You will find one that says "stopping method" or something like that (on Teco drives this is usually F09). Then, use the manual to tell you how to enter the set-up mode for the drive, and check the setting of the "stopping method" parameter. If it is set to decelerate, change it to "coast to a stop", save the settings, and put the drive back in operating mode. Then check and see if you can use the brake without tripping the over current.
Terry S.