Yes, FranMar's class I (C14 & C6 AC inlet) adapter is designed based on isolation concept, it will be no problem that the output ground (GND) and frame ground (FG) is the same point in your system. However, EMI may be affected by this connection.
When installing the power supply, both input and output ends need to be connected by wire. The following points need to be considered in selecting wires.

(1) Current rating : if the current rating is insufficient, the wire temperature will be too high or even burned

(2) Voltage drop : the wire itself has impedance, when the current flows, it will cause voltage drop. If the voltage drop is too high, it will cause the voltage at the load end to be too low, which may cause the load to fail to work.

(3) Noise : it can be a problem for wires, especially when high frequency signals travel long distances, using shielded wires may help eliminate noise and EMI disruptions.

(1) Suggest to choose a power with a rating of 30% more power than the application actual needs, in order to increase the reliability of the power supply.

(2) When using the power supply, the working temperature should measure from 1 cm next to the power supply and shall be derated if working on a higher temperature environment.

(3) Power supplies are beholden to a wide range of safety and compliance standards. These vary based on your industry, application and location. Please do make sure the selected power is qualified with safety standards and EMC regulations.