So, the energy used to propel the planet must be much more than the energy used to propel the feather.
Yes. But that is because the rocket is a special thing. See below.
It follows that the feather will traverse a distance of 1m much faster than the planet [...]
Work is not about duration. It does not depend on time or on how fast. Pushing with 1 N and moving the planet 1 m requires the same work by the force regardless of it taking 1 hour or 1 year or 1000 years.
This might not seem very intuitive at first. If it takes a long time, isn't more energy then spent since the force must be upheld longer?
The answer is no. Think of a table holding up an apple with it's normal force. No energy whatsoever is spent by the table to uphold this force. It can do this forever. Force does not require energy - apart from in specific special "machines".
And that's the thing. The root of your confusion, if I'm right. A rocket propulsion engine (most types of engines to be fair) is such a kind of machine. It takes energy to create the force it exerts. It takes fuel. Fuel is burned at a never-zero rate and so the energy consumption of a rocket does depend on time.
The human body with its contracting and expanding muscle fibres is such a kind of machine as well. Eventually you will get tired of pushing, but the wall doesn't.
When thinking of the work formula $W=\int\vec F\cdot d\vec x$, I like to compare a wall with a balloon (essentially what you are doing, but in a slighty simpler setup):
- Pushing on a wall doesn't really make any difference even when pushing hard. Nothing moves. No work is done, you are just wasting your time.
- Pushing on a balloon is easy - it moves far. But it was so easy that no one would say that you did any significant effort. We wouldn't say that you did any significant work to move it.
The first example is your situation.