I didn't check this on my JTV, but isn't a baritone tuning a perfect fifth lower? That would result in a low F tuning rather than a low G tuning.
Variax guitars create the alternative tuning by pitch shifting your strings, starting from the actual tuning (in your case, a C tuning).
I don't know how such thicker strings might effect the piezo pick-ups inside the bridge, but if it does work...
Then as E goes to B, in standard tuned strings, then C would go down to G in Baritone tuning. Baritone is not a Fifth, but a Fourth lower.
An Alto Sax is tuned open to Eb, a Baritone Sax is tuned open to Bb - a fourth down. Opposite is a fifth above.
If you are familiar with the Circle of 9 Theory, then you'll always know which ARE opposite of pitches above that equal pitches below. For example - a fifth up is equal to a fourth down. That equals 9. See?
I tested with 10 gauge strings detuned to C and super slack, but the piezo pickups still worked, although it was a bit strange sounding at times for the ultra slacked low C. I would imagine that heavier gauge strings should help with that. ALT Tuning to Baritone with the guitar like this still tracked, but again, low C was a bit iffy, but you should opt for heavier strings. You would also need to get the JTV setup and intonated properly for those heavy strings.