"Landlocked" basement bathroom with shower

a simple valve on a vent system like this would not seal against sewer gas.

plumbing code has water traps to keep out sewer gas.

Since the entire purpose of the plumbing vent is to keep the drain pipe at atmospheric pressure, attaching a fan and pressurizing the stack wouldn’t be a good idea. It would be generally prohibited as a device that obstructs or retards the flow of water, waste, or air in a drainage system.

And you’d run the risk of sewage (from your house) making it’s way into your bathroom fan.

But the ensuing chaos would make for a great Literal Photo of the Day.

Typical pitch for water runoff is 1 inch for every 4 feet. Over 20 feet, you would need 5 inches of pitch, and for 25 feet you would need 6.25 inches of pitch.

This means if your floor joists are 2x12 (which means 11 inches in reality) nominal, you will not have enough room to install a 5 inch insulated duct without ending up below the level of the joist (and thus any finished basement ceiling). Additionally, avoiding sags in this scenario is going to be difficult as well.

Drilling 3 inch holes in each floor joist over the course of 20 feet just screams BAD IDEA. There are guidelines for the hole size you can drill through dimensional lumber, you should google them. If you are using engineered lumber, it’s a completely different animal.

Besides that, a 3 inch pipe is not 3 inches inside diameter, which means you are simlpy not going to have enough flow through the tube without it sounding like a giant vacuum cleaner.

Could he use 2 parallel ducts of 3"-4" each, running through the same or parallel joist spaces, each with their own fan? This would be the equivalent a 6"-8" duct, right? Would that be workable?

If I did the math right 2 3-4" diameter ducts would have the same area as a 4.24-5.65" diameter duct respectively.

For a demonstration, cut out a square (just because it’s easier to see with a square) one inch on each side. Now cut a second one. Now cut a square 2 inches on each side. Put the 2 smaller squares into the larger square and see how much more area the 2 inch square has than just 2 one inch squares.