Adding a second button without an attic run?

So I have a house on a corner. Front door (standard transformer/indoor chime unit/outdoor button) faces one street, and a garage walk-though door faces the other street. This garage 'back' door has an old battery powered RadioShack wireless doorbell system on it, and was never wired in to the house system. I've got two shiny new Ring Pros, and I'd like to avoid running a new wire from my garage to my attic, and down the wall to the existing in-house unit. Because I'm lazy, and insulation is itchy, and the garage walls are plaster, and who's got time for all that? Wiring the garage button to a separate wall transformer would be easy-peazy. My understanding is that in that configuration, the front door would fire the relay on the existing doorbell when it's pressed, and the back door would be configured for no chime, and only trigger the apps. Would it be possible to have the garage door button fire the chime connected to the front door button when it's pressed, so that both buttons 'work' the same chime, even though one is not connected to it? I've cruised the forums a bit, and checked all the installation guides, but everything I can find says that you want to wire one to the front door lugs and one to the back door lugs on the indoor unit, so it does separate chimes for each door as intended, but if I'm ok with one chime 'melody', can I get away with my (apparently) crazy configuration?