Just a suggestion.
Your LV source may not have enough va to energize a LV contactor coil. I would suggest that you incorporate an interposing relay of which its coil, 24vac for example, will function easily with your LV source. When it is energized it would close a contact that would energize the 120v coil which controls the chime.
Also, you wouldn't be plagued with a voltage drop between the bell transformer and a contactor with a LV coil. The coil of a small interposing relay would draw very little current and be controled by a conventional door bell switch. Otherwise, a convertional doorbell switch may not be able to handle the LV current required to pull in and hold a common contactor.
I believe there would be a much less trouble using an interposing relay.
This is intended to think outside the box a bit.