My question is: what would be a good
precise antenna length to shoot for given the intended
multi-band use

Since you are using a tuner, all you need to do is insure the impedance of
the antenna system is within the matching range of the tuner at the frequencies
of interest. The simplest way is to put the antenna up and then insure that
you can tune on the bands you wish to work. If one band presents a problem,
then change the feedline length a few feet.
You could model the antenna and do an swr sweep, and that would tell you if
there were potential problems, but the tuner is a problem solver as well as
changing the line length.
The ARRL Handbook had a list of "feedline lengths to avoid" or something like
that when using a 135' dipole for all band use. As far as I know there has
never been a list for a 530' loop.
BTW, according to Reg Edwards G4FGQ above 21 MHZ your tuner will start acting
funny because of stray inductance and capacitance. I.E. the calculated values
if L and C to achieve a match will be off because of strays. I agree with Reg
on the many tuners I have built.
73 Gary N4AST