MP Products Sliding Door
I'm afraid this label doesn't tell us anything about the handle the door uses. Would it be possible to see a photo of the part in question? That would help us locate a good match. You mentioned that your mounting holes are 3-15/16", which gives me a few ideas, but there are a couple other more subtle details I want to confirm before I recommend anything for sure.
Here is a photo of the door handle
Perfect, thank you! I recommend our 82-030 handle set. I'm very confident that this will replace your old hardware.
Doesn't look like our handle, and the original is keyed. Looks more like the 080 but 3 15/16
I definitely would not recommend the 82-080. That is a very specific handle used on a specific door that you do not have. You'll note that the screws come into the 82-080 from the sides, while yours are face-mounted, which is the detail we use to differentiate.
I apologize for missing the detail about the key lock; we have the 82-086, but this has a thumb lever fixed in the bottom position and it cannot be adjusted. The thumb lever is missing from your photo so I don't know what position it's in. The 82-086 may work for you, but you need to make sure that the thumb lever is in the bottom, or else the 82-086 won't line up with your mortise lock.