Hmm, I can definitely confirm that this problem exists for me as well, and I think I've tracked down the root of it by attempting the same area in the PSX version.
The "emulate ramps" feature in FF7Music simply does not work. For the most part what that means is that every song is stuck at whatever volume it starts at when it begins playing. Unfortunately, when this song tries to resume, it starts at zero volume, with the intent of fading in (it's supposed to fade out when you go up the elevator, rather than stopping suddenly, and fade back in when you go down again). However, since it can't fade properly, it stays at zero volume until you visit an area with a different song.
Unless Ficedula shows up with a fix, we will never see this area working properly in FF7Music. The best you can do is save after going down, and then quit and reopen your game, or not to use that elevator to begin with. It's not a good solution, but it's all I've got. Sorry
To be honest, I'm surprised it's taken this many years for someone to notice and point out this problem.