The amount of your winnings is under the required amounts for Virginia, so you aren't required to file a return. You might want to file to get the withholding that was taken out, but as you mentioned, it might not be worth it if the fee is higher than the amount you had withheld.
Per the Virginia Department of Taxation:
You must file a return if:
- you are a resident of Virginia, part-year resident, or a nonresident, and
- you are required to file a federal income tax return, and
- you have Virginia adjusted gross income equal to or greater than the amounts below:
Single or married filing separately $11,950
Married filing jointly $23,900
@lsullymarl
