Replace clamp style handle with locking

A home owner from Indianapolis says:
I have an older clamp style handle that I'm trying to add a lock to. The door depth is over 2" so no lock exists that I can find. Is there a product that would adapt to the door depth so I can key the lock, OR is there an alternative handle replacement? THANK YOU!
Thumbnail of the uploaded file named PXL_20201214_194707235.jpg
Thumbnail of the uploaded file named PXL_20201214_194714794.jpg
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Tom from SWISCO responded:

Your old handle set looks like our 82-009A, which is compatible with our 82-310 key lock. This has a tail length of 2-1/4", but you didn't mention exactly how thick your door was, just that it was over 2". Would 2-1/4" suffice?

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