My garage has its own 50 amp circuit powered from the main circuit breaker panel in the house and my router is inside the house so on a different circuit.
Then there should be 3 wires going to the garage panel: Leg 1 of 120VAC, Leg 2 of 120VAC and neutral; there's 240VAC between Leg 1 and Leg 2, 120VAC between either leg and neutral.
Your router will be on the same side of the panel (same "leg') as one leg of the garage so the PLA could very likely work.
The PLA's don't require being fed by the same breaker but instead most require the 2 PLA's being on the same "leg" of 120/240 (the same side of the panel).
Actually 4 wires, neutral and ground. There are 2 hot wires from the street transformer to the house and since I have a double pole breaker leading to the garage both legs are present. So what I think you're saying is I just need to find the common hot wire from the transformer that both the router and PLA's can be connected to. So if one circuit in the garage doesn't work, (there are 4 in the garage) power the PLA on a different circuit until I find one that works.