Since the last Apple OS Update 13.4, the extension has stopped working.
It works regularly on Chrome instead.
Any suggestion on where I can head, to have this fixed?
Are you using tampermonkey with anything else? Curious if it is tampermonkey itself or the BPIP extension, I don't use tampermonkey myself so I'm not as familiar with it.
Make sure Safari updated correctly with MacOS update and isn't being packaged separately, check the version number under Safari / About Safari, then check for any potential software updates that may be lingering after 13.4 by going back to the updates area in system settings. With Apple, things usually break because of security changes where it may not trust tampermonkey and/or the BPIP extension anymore and will stop it from executing correctly.