Mortise latch
Thanks for reaching out to us. This mortise might have a bit of a strange setup. I'm not sure exactly what would install this way. I can tell you at least that nothing we have has a 3-1/2" hole spacing, and no mortise I know of installs to the back of the frame. Usually the holes are up on the face.
That said, please give us a little time to look into this. Maybe we can find something that will work in this capacity. In the meantime, please give me any other information that might be helpful. The name of the manufacturer might give us the clue we need, especially.
Oh, great! I'm glad you found a good match. Please let us know how it works out.