I searched high and low...on the 'net. And failed. Failed, that is, to find a 'mortice' lock that would fit within the thickness of the door.
Our sliding doors are, as you say, only an inch or so thick. And our cheesy, cheepie, broken Chinese mortice locks are only key-operated; there is no latch nor handles.
Plenty of residential and commercial sliding doors have 'mortice locks'; that is, they're installed in a mortice and the doors are usually 1-3/4" thick. These locks will have an associated cylinder lock (often, the cylinder lock is not the usual standard, interchangeable size). And they will have handles inside and out to operate the door.
I could have added a Teak block to the exterior face of the door and a similar block on the frame. Then the door effectively would be as thick as necessary. It might not look too bad but the door would then not slide open the whole way; the inside handle would hit the frame before the door was fully open.
Since the boat is woefully insecure with those old sliding door locks, and I cannot quite stomach the idea of an exterior hasp and padlock which would not leave us secure when aboard, I may yet add the block and lock.
Below are pics of our Norwood sliding doors which were equipped with Truth locks. These locks had "D" handles inside and out. On the outside, they limited the width of the door opening, so I made handles that gave me a couple more inches of width. Other brands of hardware use the standard size lock cylinders.