I would like to purchase a keyed lock for this handle

Sure thing! You may be able to install our 82-315 key lock and housing to your existing exterior handle. This is specifically designed to install on our 82-044, but your handle is similar and it may work just the same.

If not, however, then you will either need to purchase the 82-044 separately and use that instead, or buy a new handle set entirely. My recommendation is the 82-086.

I'm a little surprised to hear that. Looking at your photos, I don't see any reason why the 82-086 wouldn't work. Can you go into more detail? In which was was this handle set too small?

I'm still not sure I understand. Looking at your photos, your old handle set uses a single latch, as well. The 82-086 should also be fully compatible with your existing mortise lock. 

As for length, overall length doesn't really matter. What's more important is the hole center spacing.