Congress (the real, U.S. one) agreed to seat the two U.S. Senators elected by the loyalist government of Virginia, so yes indeed Congress recognized that state government as legitimate.
Sat in Richmond, faithful to Virginia's People, while her favorite son covered himself in undying glory afield with her armies.
Your fictions are so wretched, they are a joke.
Are you sure that Congress let the Senators selected by the loyalist Virginia legislature take their seats? Virginia wasn't represented later during the Presidency of Andrew Johnson. In the 1868 trial of Andrew Johnson, there were only 54 senators voting. There were then 37 states (the 36 in the Union as of Lincoln's death plus Nebraska, admitted in 1867), so there should have been 74 senators, but 10 Southern states were not represented (the 11 which had belonged to the Confederacy, minus Tennessee, readmitted in 1866).