Is GND connection required for two independent modules to communicate in MIL STD 1553 protocol?
Because the physical layer of MIL STD 1553 requires transceivers to be connected to the bus via isolation transformers, no common ground connection is required between transceivers.
The Holt IC application note AN-550 referenced in the question shows a center tap on bus-side of the isolation transformer, which it says is "often" not connected.
Data Defense Corporations Review and Rationale of MIL STD 1553 A and B has a similar schematic which shows no bus-side center tap on the isolation transformer.
If however, the bus-side center tap of the isolation transformer is grounded on one module, the differential pair will not "float", but will be referenced to that ground. If more than one module has the bus-side center tap connected to ground, and the modules are direct-connected to the bus, then there is a ground path through the differential pair. If the ground potentials are different at the different modules, this ground current could be significant, and might possibly interfere with communication. Therefore, the bus-side center tap of the isolation transformer should not be connected to ground if the modules are directly connected to the bus.
Rather than connect the bus-side isolation transformer center-tap to ground, it might make sense to connect it to the shield of the twisted pair cable. Whether or not shields should be connected on both ends has been debated. I will only say that "both ends connected" might work in some circumstances, and "one end only" might work in other circumstances. I will leave it to others to comment further on this topic if they choose to.
If the modules you are using require a common ground connection in order to communicate, they are failing to meet the standard. Something is wrong. The schematic for the Holt IC module provided in the question shows such an isolation transformer. Can you check the modules to see if they do indeed have isolation transformers? Can you verify that the bus-side center taps of the isolation transformers are not connected to ground?