Battery efficiency questions in the context of running

Would it draw more power to use the btrhm instead of the internal heart rate monitor of the bjs2 ?

'it depends' on the connection that gets negotiated with the HRM, but it's quite possible it'd be less power (as long as the on-watch HRM is turned off).

Compared to the GPS most should be negligible though.

Would it draw noticeably more power to get a bit more data out of the bthrm ?

Probably not noticeably, if there was no calculation involved.

I noticed during my runs, that the screen lights up without me touching it.

Yes, it's wake on tap. You can turn on 'wake on double tap' but even that does turn on sometimes, so if you really want to eke out your battery I'd turn it off.

Are there some enhancements, tweaks that would help

GPS is really the biggest power hog (assuming your backlight is off) but really that should last for maybe 7 hours.

It's possible the GPS could be configured differently to draw less power (eg by only receiving signals 20% of the time), but unfortunately the commands you can send to the GPS aren't well documented. Maybe if someone did a bit more digging they might be able to find something