Yes, even if killed by radiation the DNA (and mitochondrial DNA) in the donated white cells would cause a secondary genetic signature to be detectable, at least for a time after transfusion.
If I recall correctly, the spleen captures dead blood cells and breaks them down into reuseable complonents, so the dead white cells will begin to be cleared up immediately and should all be gone within a few weeks.
Red blood cells have no nucleus, or mitochondria. They suffer wear and tear from squeezing through capillaries that are
narrower than they are and wear out after about 6 months. They have a surface antigen which determines blood type. Under duress, you can be given blood of a different type to your own without a reaction, so the recipient could have two detectably different types in circulation for some months.
However, the laboratory determination of type, during that overlap, would be problematic (but I don't know the details myself).