Actually, the NPS might be savvier than you think. Transportation-related federal grants are notoriously inadequate for large projects. You have a $250m project and a grant gets you $25m or some other relative pittance (more frequently, some schmoe of a congressperson gets you a $2m earmark and wonders why the project never got off the ground). Someone's still got to pony up the rest of that money, so there's no reason to apply for a grant that will sit around and lapse anyway due to lack of funds.
The Arlington Memorial Bridge, contrary to what many people beyond the reach of this forum believe, does not reach Virginia. It is entirely in D.C. (though it obviously serves many Virginia drivers, and it used to serve many buses based in Virginia, but not since the bridge was posted with a 10 ton weight limit).
Because it is entirely in D.C., it may be difficult to get a bill through Congress to fund the repair of the bridge (though it is obviously a federal project, and the bridge has always been owned by the federal government) - since D.C. has no voting representation in Congress.
IMO, this would be appropriately handled by a separate bill and funded by a special ["pork barrel"] appropriation.
If I had
my way, the entire federal NPS parkway system in the D.C. area would be tolled, and the revenue collected from those tolls would first go to maintaining and improving the parkways and related infrastructure.