My method doesn't require any additional bits for the e.p. and is compatible with all the other techniques listed in TFA.

Using own king's position loses the info of the file. Iff you guarantee that the pawns are listed in order, this is not a problem. But later in TFA a permutation or sorting of pawns is suggested, which would then mess things up.

Please note that a slightly modified version of my method, where a pawn that can be captured e.p. is swapped (not merely moved) with whatever is in its corresponding back rank before any other encoding takes place, is compatible with all the other clever techniques suggested in the comments.