Keyed access for aluminum sliding door
I removed the handle and here's what I measured. The mounting holes are actually 3-1/2". The lock lever is below center. I looked on the site and couldn't find any sets to match those specs. Do you think I could use a 3-15/16" set and just drill a top hole to make it fit?
I think your door handle may be a bit too old for us to find an exact match for. Regardless of what we recommend, you'll almost certainly have to redrill new holes.
One thing I'm concerned about is what that thumb lever is turning. It's definitely turning a mortise lock of some kind, but based on how old your handle set is there's a good chance that your mortise will be too old to replace, too. I also can't judge if our standard 3-15/16" handle sets will be compatible with that mortise. Seeing a photo of yours out of the door would be a big help.
Another option to consider is our 82-005. This a surface mount handle set, which means it uses its own self-contained mortise without having to interact with anything in the door. This allows it to install on just about anything, though drilling holes is necessary. This might be a good last ditch emergency option, should all else fail.