Having established that simply replacing the beveled washer wasn’t going to work, I examined the marks left by the valve seat on the washer. Obviously, it wasn’t quite sealing all the way around the seat’s inner diameter: much of the washer had an indentation, but about 1/4 of the perimeter was unmarked.
I couldn’t determine if the valve seat was flat or beveled by looking into the partially water-filled body, but I figured that if I could burnish the edge of the ID to make it more even, then whatever was left should seal better against the washer. Measuring the included angle of several different beveled washers showed each size has its own angle, so I turned the end of a steel rod to match the washer installed on the faucet shaft:
With that in hand, I filed some shallow radial indentations on the bevel, eased it into the faucet body against the seat, and gave it a few turns while pushing firmly. The intention: smooth the existing brass, not cut a new seat.
Although the plan made perfect sense, the faucet leaked much more vigorously with the valve cranked closed.
In retrospect, I should have tried a stack of flat washers in hopes that they would seal across the entire valve seat. However, the screw in the end of the valve stem reached the end of its tapped hole at the right distance for a beveled washer, leaving far more space than any single flat washer could fill, so this obviously wasn’t the right way to go. Here’s the screw again, seated firmly at the bottom of its hole:
At this point, I had to decide whether to continue futzing with the valve or chisel it out of the wall and replace it. Time for a trip to the Big Box Home Repair store to see what’s in stock… which turned out to be an assortment of frostproof valves in 2 inch increments from 6 inches to 12 inches, with the latter about $28. The quality appeared to be marginal, the designs included fragile plastic bits (some of which were already broken in the shipping bags), and it was not obvious that they’d outlast the gardening season. The online reviews were, shall we say, equivocal.