Not being a GS, but a lowly 1k (although, we all know that sometimes 1k>GS in the eyes of UA) - I am somewhat surprised (a little) that a call to GS CS resource that I assume you have access to - and explaining the story/asking the question would result in an absolute "no"
- they might not have a cheaper fare bucket available, but maybe there is something else that someone could do that would get your $ in exchange for a lay flat TCON seat

I know that I have called 1K CS a few times over the years and have gotten help with issues in which I was extremely surprised about the outcome in my favor - to the point in which I was even afraid to thank someone out of concern of getting them in trouble.

maybe it also depends if "you" (i.e. OP) are a GS with $100k yearly spend or a well-placed exec at Apple (for example) who flies a handful of times a year but plays golf with Tim