Upgrade suggestions from BAT VK-300xSE

I'm looking for advice on an upgrade from my BAT VK-300xSE, without breaking the bank. I'm considering Audio Research LS25MKII & 300.2 or VT100MKIII (used). Or BAT separates, but I've been told; that I'd have to go way up the line to better their Integrated by a significant margin.
My Speakers are Wilson Audio Sophia.
My Source is Krell SACD StandardV.2
I agree, your vk300se is pretty darn good. I am not sure the audio research gear you list "significantly" betters what you have. Different yes. Good yes, better???

What's your budget? What cables are you using?
I went from a 300X to the VK-31 preamp and VK-200 amp, then the VK-500 amp, and heard significant improvements across the board with my demanding speakers, especially in the bass. (The power supply in the 300X and 300Xse is limited by the amount of space in the chassis.) If you like the BAT sound, you could check the price of used SE-type preamps and one of the BAT solid-state amps, and compare these prices with those of comparable ARC units.
I had Sophias, now WP7's, and ran them for a while with the BAT VK-500, then VK-600. Great bass and dynamics but no soundstage. My recommendation would be an ARC VT100 Mark III or Conrad-Johnson Premier 140. You will lose a little on the bottom end but gain so much soundstage size and improved tonal balance that you will wonder why you ever tried the BAT solid-state equipment. Or if you ever see some used Essence solid-state amps, they are the best I have ever heard with Wilsons. I am not sure what you are using for a preamp, but a tubed job like CAT or BAT or ARC will also greatly improve the sound.
I'm a 300X-SE user who formerly owned an ARC VT100 Mk-I &
I'm my circumstances (smaller room, lower volume listening) I much prefer the BAT to the ARC.
The tubes generally sound "polite" and homogenized -lacking in color and texture.
I swapped out the ARC for some borrowed BEL 1001 Mk-V monos ($8000 pr.) and was stunned by the improvement!
I guess I'm just a solid-state guy at heart.
I'm leaning towards the BAT VK-31SE (good deals on out-going model) & BAT VK-250, which I will demo. this week, though it won't be cheap at $12K-retail (2X the cost of my VK-300xSE). I'd rather not mix-and-match pre-amps & amps. Manufacturers of this caliber go through a painstaking process when voicing (tuning) these components for optimal mating, and I doubt that I could better. The ARC LS25MKII & ARC VT100MKIII are my second choice, at around $5.5K-used, though I’m hesitant to take on the bulk, heat & maintenance of a very large tube amplifier.
i ran a 300xse with my sophias...then stepped up to 31SE/75SE. whole different league at that point. 51SE was an even greater improvement.

good luck!
The last time I checked, a VT-100 factory retube (power tubes only) from a dealer costs $850.
This is every two years if amp used daily (1-2 hours) and weekends (4-6 hours).
You can use non-factory tubes, which go for around $400,
and bias yourself.
This is partly why I switched to SS.