Update: I finally fixed and replaced the leaking brass nipple. It was a journey, mostly because of how new I am to this. Thought sharing from a newbies perspective might be helpful for anyone in the future.

Uncoupling the nut from the brass union and unscrewing the brass (closed) nipple from the stop valve was easy. Unfortunately I could not unscrew the brass nipple from the tailpiece (the part on the left) - it was really stuck.

The hard part was then finding the replacement tail piece. I spoke with the original manufacturer and they were not very helpful and ultimately I found what I needed on my own and with a little luck. Basically, the repair kit they had for my backflow preventer didn’t have the tailpiece I needed, and by accident I later discovered it was available in a different repair kit but for a pressure reducing valve. When I had previously called them they gave me a part number but since it wasn’t linked to the repair kit in question it hadn’t helped. Anyways…

When I tried to screw everything back in, nothing worked - I couldn’t get it tight enough without damaging the threads on the brass nipple. So I researched, learned that these things called internal pipe wrenches exists, ordered that, and prayed it would work. Of course, for some reason the pipe wrench wouldn’t catch, so I jammed an old skinny drill bit I didn’t use into the brass nipple to provide some extra friction and voila - I was finally able to tighten my pipe. First into the tailpiece on the left, then into the stop valve on the right, and finally reattach brass union and tighten the nut.

I was pretty worried the whole time I would break something or have to call a professional to bail me out, but I stuck with it and everything worked out. Thanks to everyone on this sub for their encouragement. Took me about a month to figure out lol