xp_fixeddrives is an undocumented procedure. So basically, it doesn't exists. the is no documented way to grant, deny or revoke any permission on it and the result of executing it can be basically anything.
That being said, you can try to wrap the XP in an ordinary procedure, then use code signing to grant the needed permissions to the procedure, then grant EXEC to your user on the wrapper, similar to the example in my blog: Signing an activated procedure. This works on my machine. Your mileage may vary, as is the case anytime you use undocumented functionality:
create procedure usp_fixeddrives
with execute as caller
grant execute on usp_fixeddrives to [low_priviledged_user];
create certificate [usp_fixeddrives]
encryption by password = 'AnyPassword@1234!'
with subject = N'usp_fixeddrives'
, start_date = '11/05/2009';
add signature to [usp_fixeddrives]
by certificate [usp_fixeddrives]
with password ='AnyPassword@1234!';
create login [usp_fixeddrives] from certificate [usp_fixeddrives];
grant authenticate server to [usp_fixeddrives];
grant control server to [usp_fixeddrives];
alter certificate [usp_fixeddrives] remove private key;