Trigger two circuits with single SPST switch
Wire a relay that can be triggered with 6V in parallel to your chime. That relay will look like a closed switch when the coil is energized.

By the way, your 2× 3V battery setup is very strange; since the permutation of devices (switch, batteries, chime) in this setup doesn't matter at all, as long as all stay in the same direction, you'd normally have one set of batteries in one place, rather than distributed to both "sides" of the chime. When you rearrange that, you can see that you can replace your two 3 V voltage sources by a single 6 V voltage source – which, in the case of a door bell, can usually just be a cheap "wall wart" power supply, which will probably be easier than replacing the batteries occasionally.