Install door handles on opposite sides of a pocket door

I am new to DIY and bought some custom door handles; I’d like to install them on both sides of a pocket door, but I’m not sure how. I asked the manufacturer for an installation guide, but they haven’t gotten back to me.

The handle has two holes, one for a 1/8" screw and one for a very small metal dowel, about 1/32" wide and 3/8" long.

The 1/8" screw is 1 7/8" long and goes in 1/4" deep into the handle. It has a cap nut that covers 1/4" of the other end, and a washer that is 1/32" wide.

The 1/32" dowel is 3/8" long and goes in 1/8" into the handle.

My door is 1 3/4" thick.

I’d like to install handles at the same height and mirrored orientation on both sides of the door; it seems like I could do that by drilling a single hole through the door and using a dowel screw 1/4" + 1 3/4" + 1/4" = 2 1/4" long, but I’m not sure that would work. On the other hand, I don’t want to have the fastener for the screw showing on the opposite side of the door, which would also prevent installation in a mirrored conformation.

Pictures of the assembly below: