Converting this to a locking handle

Thanks for reaching out! The 82-306 Keyed Exterior Sliding Door Pull should work to convert this door to a keyed lock. Like the existing set, this can fit a 3-15/16" hole spacing.

If you need to replace the entire set and would like to stick with the same style, I suggest also purchasing the 82-020 Mortise Style Handle Set. Another option is the 82-030A Centered Keyed Handle Set, it is made in a variety of finishes.

The mortise lock in the door reminds me of the 82-226 Mortise lock. It is sold separately from each of these sets. Be sure to double check its dimensions to confirm it is a fit. They can be viewed on the website.