HM10 bluetooth module to discoverable state after connection break
I've worked on a project that involved an HM10 and experienced similar issues. Here are some ideas for debugging the issue:

Check the System LED function (PIO1) pin with a multimeter to determine if the HM10 is in a connected or disconnected state (see datasheet).

If it is in a connected state, that would explain why it is not emitting BLE advertisements. But if you physically remove it from any other BLE devices it can pair with and it remains in a connected status... this is strange.

What you may want to do next is check whether sending a reset signal to the HM10 will fix the issue. From what I understand, having the 328p sent AT+RESET may not work, as the docs state the way to sent AT commands to the device while it is in a connected state is from the remote device, and when the HM10 is in the remote control system work mode (AT+MODE2).

You can try sending a reset signal through the System KEY function (PIO0) by connecting it to ground for > 1sec. You can also try this for the RESETB (11) pin - I've found this one to be more reliable than PIO0. If this works, then you would just need to connect this pin to an output pin on the 328p and send a LOW signal to the HM10.