OP please don’t purchase a Kwikset to replace this. Look into the Baldwin Estate line of products, I don’t know about current production but they had similar operating locks with the “push pin” style setup. You will need to mortise the faceplate those use a weird oval that will not match up anything current. Likely you will also need to redrill the mounting area for the new lock because your existing hole probably won’t work with a new lock of any type.
Unless you have the woodworking skill you would be best off ordering a nice privacy lock with a knob/lever that matches your current other locks and hiring a locksmith after making sure they have the ability to actually do the work correctly