They shouldn't need fingerprints to do a background check.
They use those prints for two purposes.
First, to see if you are already in the databases
(for example, are you using an alias?).
Second, they keep them on file so that when
something bad happens in proximity to what
you are being cleared for, they can see if
your fingerprints are all over it.
The actual background check is only the smallest
and beginning of the many things you are consenting to
when you apply for a clearance.
I (and my fingerprints) are in so many federal
and state databases that I could never disappear.
(Well, unless I had all my fingers mutilated in a
very special way.) Security clearances, concealed
weapons permits, teaching (ie. children), etc.etc.
Many times and varieties over more than 60 years.
I don't worry about it at all.
But then, someone concerned about that sort of
thing would never do all the many things I've done.