Need to Add Outside Keyed Lock to Sliding Glass Door Handle

I have a sliding glass door with a handle that appears to be your 82-005 with a 4 15/16 screw spread, installed new when my house was built in 2000. I want to add a keyed lock on the outside. The 8 1/4 inch long plastic pull on the outside is fully recessed into a 7 7/8 x 13/16 inch cutout on the outside of the door. The pull appears to have a scored circle on the inside and a structure for adding a lock. This structure is similar to a universal retrofit keyed door lock/pull kit I purchased but does not quite work. And if I could get it to work the outside handle of this retrofit kit protrudes so much the sliding screen would not have clearance. Can a lock be added to my existing recessed plastic pull, and do you sell that lock? Since this circle is scored only from the inside, would I drill it out for the lock face to pass through without harming the pull, or should it be punched from the inside? Do you have a replacement part for this pull in case I try to drill or punch it out for a lock but experience a problem? I have attached photos for reference. Thanks for your help.

Thanks for the advice. I will try this with a slow-speed drill. In case I have a problem, is there a different handle set available with an outside pull that will be compatible with the existing 7 7/8 x 13/16 inch cutout in the door?