Installing a lock where a plain handle existed

Hi. My sliding glass door has handles on the inside and the outside. These are however, dummy handles. There are two screws that go through the door, positioned vertically one 5.5 inches higher than the other. I cannot however, tell if there is any hollowness between the two handles. I think probably not, because there is no opening for a latch to come out on the end of the door. So, I assume that the entire end of the door, including the space between the dummy handles is probably filled with a wood frame. Assuming I could cut a notch in the end for the hook latch, and a small hole on the outside for the key part of the handle would go, which would be the best (? smallest) of your keyed handles that would involve the lest amount of cutting. I'm comfortable with Dremel type tools so I am confident I can cut through what appears to be an aluminum casing and that I can bore out the wood safely. The door is 1 and 3/4 inches wide. Looking at the door end on, the center of that 1 3/4 inch door would appear to be 1 and 1/8th inch. So, my assessment is that there is a wooden core around that is 1 and 1/8 inches thick. There is also plenty of space from the end of the door to the glass, so it should be safe to cut out a void and not hit the glass. Which model would you recommend? (Also, the 5.5. inch inter-screw spacing isn't critical as I can just drill an additional hole if needed.) Thank you.