Could bypass air be used to cool a rocket engine like in turbofans? Rocket engines typically employ a cooling jacket of sorts, usually dual-purposing the fuel as a liquid coolant before it flows into the injectors.

Here is an example turbojet using a bypass. As you can see, the bypass air is far cooler in temperature than the air combustion chamber:

Source

Why are there no designs that use an air bypass (similar to a turbojet) to cool the engine/nozzle? Obviously once the rocket has escaped the atmosphere, a bypass would be of little use. However usually by that stage in a rocket, thrust has been greatly reduced and therefore generated heat as well.

Wouldn't a bypass cooling system make a rocket engine a tad less complex (especially for the first stage)?
Air cooling is not going to contribute much here.

The Shuttle engines were cooled by liquid hydrogen (-253°C) from the fuel supply being pumped up cooling ducts inside the nozzle. Typical nozzle temperature in operation was around 54°C. Yes, fifty-four Celsius. (BBC Engineering Connections - see this Youtube clip)

The top speed of the SR-71 (about Mach 3.2) was limited by the temperature of the air arriving at the engines which, after being compressed in the inlet ducts, reached temperatures around 400°C. (Wikipedia: SR-71)

The shuttle passes Mach 4 about 2 minutes after launch.