Integrating wireless transmitter with old door chime system
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. What you've probably got.

There's something strange about your setup. If there is a short where the bell was (LO-Z in Figure 1) then there is a danger of the transformer overheating if the button is held in or fails short-circuit. If the bell had simply been removed there would be two breaks in the circuit and you wouldn't detect voltage on the switch - unless it's a high impedance meter picking up strays. As per my comment, see if there is enough current to light a 6 W, 0.5 A car tail lamp.







simulate this circuit

Figure 2. AC-less solution.

Since you don't seem to mind using a battery, the AC is no advantage. It can be disconnected and the house made a little safer. You'll need to figure out which button has the supply.

If you purchase a remote control coded key-fob or a standard wireless doorbell system you should be able to solder some wires onto the push button pads or leads. Extend these out and hook up to the parallel switches. I really don't know if the long house wiring will affect the input so some testing may be required. A small capacitor - 100 nF? - across the original fob button may do the trick.