Which would be why all that needs to happen is for Congress to appropriate exactly 1 penny to the renovation of the ballroom. There's your statutory authorization.

Well, that only authorizes the Administration to spend one penny. Per the judge's order on the preliminary injunction, the Administration also lacks any statutory authority allowing it to accept outside funds and direct them to the project:

The President may at any time go to Congress to obtain express authority to construct a ballroom and to do so with private funds. Indeed, Congress may even choose to appropriate funds for the ballroom, or at least decide that some other funding scheme is acceptable.

cdn.savingplaces.org: PDF document

That's in the conclusion, but it's of a piece with the substantive discussion elsewhere in the opinion that Congress has not authorized the President to accept private donations to fund the ballroom, and that such authorization would be necessary for him to be able to do that.

Perhaps that's not all that much of a difference, because I suspect that any bill that could get through Congress containing the former could also include the latter. But I don't think just adding a penny in appropriations alone would do it.