I don't know if this would work well for your player, since it kind of assumes that the construct fights instead of the main character, but you could handle the construct like shape shifting and create the construct with the same number of skill points and the same skill cap as the character. So outside of combat, the player uses his main character's character sheet. Then in combat, he switches over and uses the constructs skills.
In most ways, it would be treated as as single character except for having two different skill builds. The aspects and fate points would be shared (after all, it's the main character who's the star and has free will, while the construct is, well, a construct).
If the character wanted the construct to have Powers, like strength, toughness, etc. (which seems likely), they'd have to be purchased by the character, so it would probably work best if the construct was his main schtick and he wasn't also, say, a wizard. (Though I suppose something like modular powers might work, with the main characte and the construct having the same number of refresh spent on Powers, thought the Powers are different between the two forms.)
Aspects would be tied to the main character. The character could and should take at least one aspect tied to the construct. This would help keep both characters narratively in the scene, even if only one is being played as a full character at a time. Aspects about being protected by the construct could be compelled to have the currently background character (whether the main character or the construct at that time) cause problems, be missing, be threatened, etc. And they could be invoked to have the supporting character help out.
Like I said, not sure if it would do what the player wants, but it seems like it would be easy to do mechanically and it could be fun to play. But it does sort of require that the construct guardian be the character's main schtick.
-John B.