Bill,
The BAT presentation is very different than that of the ARC. It's not all about refinements of tonal coherency, stronger frequency extremes, resolution, but more so about whether you like the presentation much more projected out into the room or more so back at the plane of the speakers.
Before you jump at the 31se, there are many worthy tube line stages in this price range. I assume you have a fully balanced system and thus you want to stick with a balanced line stage. This eliminates many great products. But the Aesthetix Calypso fits the bill as a contender to the 31se. There are definite pros/cons of each. Moving up to the Aesthetix Callisto significantly beats each of these at their own game but at a much higher price.
You may or may not like the BAT sound. Make sure you get a chance to hear it in your own system before you commit. For me the 51se was a refinement over the 31se as much as the 31se was over the LS5 III. And so I did not bother with the 51se. As much as I liked the 31se, I quickly found myself wanting line stage performance found at the next level.
If you want to skip all these incremental improvements, you're into the $5-7k range with products like the Aesthetix Callisto, Aria WV, Arc Ref3, VTL 6.5/7.5. The VAC Rennaissance or Lamm L2 are serious contenders but these are single-ended.
John