How should this chord be named, and does it include the B in the melody? Am I even spelling this chord correctly? (Side question: Am I resolving this correctly?)
I interpret it as a G#dim chord, because it contains all three notes of that chord and resolves to Amaj. In that interpretation, the B# is a non-chord tone, acting as chromatic lower neighbor to the C#. All of the pitches resolve correctly.
There isn't another formation for the chord that neatly fits into the stack-of-thirds approach to chords.
CMaj7(#5). (B# seems to be an unnecessary enharmonic spelling if you were notating the chord on the sheet, since CMaj7 is in the key of G).
Yes, this is a working chord progression (I'm also hearing the 2nd chord with some alteration). G - C#7(#11) - CMaj7(#5) - AMaj/C#. It feels like you could also resolve to A minor first, and then in the fifth measure go to the A Major.
The really weird note to me is the G# in the second measure. But I hear how you have it as an inner moving voice (G-G#-G#-A).
I think notation (for the purpose of other people playing the music) comes down to one of two things: 1. Do you want to be theoretically/enharmonically accurate or 2. Do you want it to be readable so that people can play it accurately?
I think the B# is actually a pretty good choice for readability on the staff in that context. But maybe a C natural would like fine too. As far as labeling the chord, I would call it a C.
Hope that is helpful.