I own a BAT VK5 (not for sale) that I upgraded to the 5i years ago. I also had Upscale Audio do a tube upgrade. Despite these upgrades, as I heard other gear as time moved on, I was not totally satisfied with the sound, and back then the Audio Research offerings were better (IMO). The VK5i also runs way to warm (hot), but not sure about your VK3. This may be due to the tubes getting over-cooked.
I also love tubes, but I realized the importance of system matching your components. I wound up with a CJ Premier 11a amp back then, because I could not quite afford the BAT VK60 mono blocks. I always felt, that this combo never really worked that well together and was holding my sound back.
After this experience and moving forward, I worked diligently to system match my components, but had a fairly recent surprise. When I replaced an integrated amp that was better than my old CJ/BAT combo with my new 300b mono blocks, I found myself without a line stage. So, I inserted my 1990 Boulder L3AE line stage into the mix and got lucky. My system transformed into the best sound I have ever had and I can now hear anything I change. This doesn't mean that the Boulder won't get replaced, but I have at least gotten to a resting place where I trust my sound and gear.
My point, when you're trying to choose a new/used component, is to try and make sure that it will mate well with what you've got and keep building out your system with that in mind. Avoid, apples and oranges.
Kenny