I’ll too be afraid of voltage loss over the vast distance. I would install Romex 14/2 from the first transformer and tie it into the 120VAC. But switch it with the output of the first transformer. Use a relay from the first transformer to power the 14/2. Then just wire the output of the downstream transformers directly to the chimes.

When the door switch is pressed. It will chime the first one and the others at the same time.

KOT