Is there a way to adjust the Bluetooth transmitting power? Either via the system or some 3rd-party app?

I doubt, i suppose that is regulated in the firmware in the radio itself.

I might be wrong, this class allows changing tx power on android: https://developer.android.com/reference/android/bluetooth/le/AdvertiseSettings.html

However i assume that is to change between the min and max value of tx power of your radio for bluetooth. If in your original question u meant to go beyond the max tx power, i'm afraid that won't be possible.

Not specifically related to phone, but in my opinion, interesting read regarding tx power of bluetooth:
https://docs.silabs.com/bluetooth/5.0/bluetooth-general-system-and-performance/bluetooth-tx-power-settings

I'm not sure if there is an app for it.