My winter system is a BAT VK55 with ESP speakers. I change speakers and amp for summer use. Summer is W4S ST500 with Nautilus 802's. The sound is different for sure but I really enjoy both systems but for different reasons. With the N802's the ST500 is superior to the Classe amps I had prior. The ST500 sounds best in my system with an Audioquest NRG10 power cord.
Therefore you might consider keeping your beloved BAT for use in the cooler months and adding a class D amp for summer. I'm sure you can find something reasonable in price to try. Then you will know for sure how they compare. I purchased the ST500 new from Underwood Hi Fi for around 1600.00 and I'm very pleased with it.
My preamp is tubed (VAC) and all interconnects and speaker cables are the same between systems. I do change power cords and I alter the room treatment adding tube traps to the front wall corners due to the N802 ported bass. So minor changes between the BAT and new class D amp may be required.
The efficiency of class D is certainly welcome for this desert dweller.