Would building a deep-bored tunnel across the Potomac work? I know it would be expensive, but I don't see them building any new above-ground crossings over the Potomac.
Tunneling under the Potomac wouldn't solve the problem, unless maybe if it came with very long (about a dozen miles) and expensive tunnels for the approach roads (say, between VA 7 and I-270), to avoid residents anywhere near the river who don't want a freeway impacting their neighborhoods.
I seem to recall that was one of the reasons a Woodrow Wilson Tunnel did not advance as a proposal for replacing the old Wilson Bridge–the tunnel would have had to have been far too long on both sides of the river in order to allow for a sufficiently gentle grade to keep traffic moving at a reasonable speed (primarily due to truck traffic), and the problem would have been exacerbated on the Maryland side due to the slope immediately east of the river. The same issue was cited as one of a number of reasons for rejecting a high-level bridge that would have eliminated the need for a drawbridge–I read somewhere that the uphill grade for a high-level bridge would have had to have started near the Telegraph Road interchange in Virginia and would have exponentially increased the cost of the bridge (not to mention the visual impact to the historic areas in Old Town).
The other issue with a high bridge over the Potomac River between Alexandria and Oxon Hill beyond the long "uphill" climb on the Virginia approach was that such structure would have really been a hulking presence over Old Town Alexandria, since the navigation channel is near the Alexandria shore.
As a comparison, consider that the WPL (Chesapeake Bay) Bridge has 186 feet (57 meters) of clearance over mean sea level between the suspension towers, and that most of the buildings in Old Town are maybe four stories tall (this is not Condo Canyon), though the Coast Guard requires 135 feet (about 41 m) on the Potomac River, so a high bridge would not be quite as high as the WPL.
But by building a draw span, that made the height requirement much, much lower. I am sure that the FAA was quite happy with a lower span too, given the amount or traffic into DCA most days.