Err...
I'm afraid I don't quite grasp some of this.
First, I'll presume that the "boiler" is the same vessel as the "kettle". I then assume that you are boiling the wort in your "kettle", then taking it from the shed to the kitchen, where you are cooling it with a wort chiller before transferring it to FV.
Anyway, to answer your specific question then, yes, I see no reason why you cannot cool the wort in the FV. Assuming that the FV is a standard fermenting bucket made of HDPE then it will have no problem receiving wort at boil temperature (although I wouldn't move it, as the plastic will be softened & might buckle). You can cool the wort in the FV, preferably aerate it, then pitch the yeast.
But this answer doesn't seem quite right to solve your question.
Why not either:
1. Boil in your kitchen, and cool there.
Or
2. Run a hose from an outside tap (or wherever feasible) to your shed, so again you can boil & cool in the same place
Or
3. Don't bother chilling the wort. Let it cool naturally. (Warning: lots of people do this - but it has significant implications for the bitterness of your beer. Recipes that require a rapid chill do not transfer to no-chill easily if there are late additions of high-bittering hops)