Most $13K speakers will easily outperform those Harbeths IME, even with an entry-level integrated, so I believe you’d be much better off upgrading the speakers with that budget. There are ≈$13K speakers (and much less with DIY options) that do everything better than a pair of 30.2s, even midrange and vocals.
I do however agree with @mbmi that Audio Group Denmark (Aavik, Axxess) make some excellent integrateds in this price range that would likely have good synergy with Harbeths. I have a Forté 1 that I preferred to both the $9K Luxman 507Z and a Bel Canto E1X in a back-to-back comparison. It is a rarity of a SS amp in that its largest distortion harmonic is 2nd order.