Can This Become a Keyed Lock Without Replacing Parts?

Good morning. I am wondering if we are able to install a keyed lock cylinder into this current setup instead of replacing the handle itself? I have the outside and inside handle. My customer did not supply the locking bolt/hook itself. Will an 82-310 lock cylinder fit in this current setup or would we need to order an 82-306 to replace the outside handle? The outside handle seems similiar to an 82-306 but with different dimensions and screw holes.

I also have more pictures and measurements if that is helpful.

Thanks,

Nate