Okay. It was unfortunate that the Jefferson Co marriage record only listed the place of birth and not the parents.
I would suggest that you look at the Prince William County Personal Property Tax Lists for the early 1800's. Those should tell you which Beavers are living there at the time. I have not seen those lists, but if you are lucky and the enumerators listed the names of all the tithable males in the household, Alexander would appear in his father's household after he was 16 years old, assuming he grew up there. Most counties, however, did not name all the tithables. Those lists are available through the Inter-library Loan department of the Library of Virginia and can be borrowed through your local public library.
Pr Wm Co Reel 288: 1782-1810 Personal Property Tax Lists
Pr Wm Co Reel 289: 1811-1832 Personal Property Tax Lists