I want to replace my wired doorbell but keep my illuminated wired push button. Can I change to a wire-free doorbell?

Sure. Just use a Honeywell converter (Part number RPWL4045A). It connects to your wired doorbell push button and bell transformer, allowing it to work with Honeywell wire-free doorbells with ActivLink and/or "E" Compatibility (916.8MHz).