The input impedance of the Halo A21 is spec'd at 33K unbalanced and 66K balanced, which would be a suitable match for most tube preamps. I'd still be hesitant to pair it with the BAT preamp, however, assuming the output impedance of the VK-3i is similar to that of the VK-3ix. Although with recordings which don't have significant content in or near the bottom octave you may not notice any issues.
Regarding use of a tube power amp with your speakers, I couldn't find an impedance vs. frequency curve for the 803S. However, impedance curves for several other B&W models, including some that seem to have a degree of commonality with the design of the 803S, all show a rise to very high impedance values (well above 20 ohms) in the upper midrange/lower treble area, around 2 to 4 kHz or so.
The interaction of that kind of impedance variation with the relatively high output impedance of a tube amp will result in greater emphasis of those particular frequencies, relative to other frequencies, in comparison with the frequency response that would occur with a solid state amp. That would seem to be the opposite of what you are trying to accomplish, namely softening the highs.
As others have indicated above, when it comes to power amplification B&W's tend not to be tube-friendly.
Regards,
-- Al
