You don't actually need two transformers to achieve isolation. The one step-down is enough; its secondary is already isolated from the primary.
Is there a safety issue with having a separate ('dedicated') GND distinct from safety GND? No, that's perfectly ok. That's typically the case with electronics. Phone chargers for example have a separated secondary ground.
One drawback to having the separate GND is that some leakage noise can develop between it and safety GND. This leakage is limited in current, but you can still measure it as a voltage between safety GND, sometimes as much as 60V or so. This can cause problems sometimes, such as hum or noise pickup (example: IR receivers) or other sensitive circuitry. Adding a tie to earth GND can mitigate this.
By the way, if you're using a full-wave (4 diode) bridge, you can't connect anything from the transformer secondary to GND. Connect GND only at the DC side of the rectifier. If you're using a half-wave (2 diode) rectifier with a center-tap secondary, tie the transformer center tap to GND. As below (simulate it here):
