I have used many amps on the Harbeth SHL5 before, currently own the SHL5+ and Luxman L-590AXII, Sonneteer Orton and Naim NAC 282/NAP 250DR. The Naim sounded best with the Harbeth to my ears and preferences. The Harbeth sounds a bit dull, flat and uninspiring with all other amps, including the Luxman. Having said that, I noticed there are few people who matched the Harbeth with Luxman without any issues.
I do not have experience with Unison Unico, LTA or other tube amps with the Harbeth.
FWIW, the Luxman sounds wonderful when matched with Marten Dukes in my main system. In this system, the Naim is outclassed by the Luxman. Synergy (and/or maybe listening preferences) is key.
Good luck with the amp search.