Most Cougar trailers have a FloJet water pump, not a ShurFlo pump. They perform the same function, but are not "readily interchangeable" because the terminal fittings for FloJet pumps use an O-ring/locking lever and ShurFlo uses threaded connectors. To change to a different brand pump, you'll need to also change out the pump connection fittings on the trailer plumbing lines.
That said, I'd suspect your pump is a FloJet. The internal pump head diaphragm serves as the check valve. Any debris or trash that gets into the pump head will prevent the diaphragm from sealing well and will cause "backflow into the fresh water tank" when connected to city water. You'll also hear the water pump cycle off/on/off when on internal tank water because as the water "bleeds back through the pump head" the pressure goes down causing the pump pressure switch to activate to raise the water pressure.
During trailer assembly, any sawdust or plastic drill shards that fall into the fresh water tank can/will get drawn up the supply hose and get trapped in the pump head. That will cause exactly what you're experiencing.
Taking the pump apart to clean out the trash is an easy project for most people. Just pay attention to how you take it apart so you can reassemble it the same way. MISSING A STEP CAN CAUSE THE PUMP FAILURE... and you do not want "extra parts when the job is done".....
The way to prevent this kind of failure is to install a "water strainer" in the supply line from the pump. They are available for FloJet and ShurFlo pumps and are not interchangeable.
This is a photo of the FloJet strainer. They are available at Amazon for less than $10 and are one of the "first modifications" I'd recommend for anyone buying a travel trailer. The strainer is serviceable, so every year, remove the screen and clean it.
https://www.amazon.com/Flojet-017403...s%2C214&sr=8-3