When attempting to reconnect, the DS4 first tries to connect in "ps4 native mode". Normally this is rejected by windows bluetooth, which makes the controller tries to the connect in "PC mode". With the BthPS3, the DS4 actually can connect in "ps4 native mode", but windows doesn't know what to do with it. Since nothing actually picks the controller data, BthPS3 drops the controller after 10s by design.
To "fix" this, Nefarius implemented a function that instantly drops the DS4 controller when BthPS3 detects that it is, well, a DS4 controller, then disables itself for 10s, so that when you try to the connect the controller in this 10s window it will "sucessfully fail" to connect in Native mode (since BthPS3 is disabled for 10s) and then it will fallback to PC mode, succefully connecting.
Open Windows' Regedit then go to -> HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\BthPS3\Parameters
After clicking in "parameters", on the right side double click on "isWIRELESSSupported" and change from "1" (Yes, it is supported) to "0" (No, it is not).
After that, close regedit, the try to connect your ds4 controller. It should connect but then be instantly dropped, after that you will have 10s to turn on the controller again for it to connect normally.
EDIT: I sent a message in Nefarius' Discord server warning Nefarius that this is happening. From what I read on the FAQ, the default parameter for "isWIRELESSSupported" is supposed to be "0" by default. Probably he will fix it the update the setup