I will jump into this one, since we're in the Dead Time and I don't have games to get anal over.
Splits clearly impact BA on a nearly-linear scale (until you get to really low splits). Guys with 80-80 splits will have a predictable delta in BA over guys with 60-60 splits, and those guys will have a similar advantage over guys with 40-40 splits. Of course, the vR split is the more important due to the greater amount of RHP, but vL clearly has a BA impact. At some point, declining splits seem to have a bit less impact on BA. A guy with 0-0 Splits won't hit .000. I had a starting SS for several years (because he was cheap) who had an 86-84-97-85 glove and a weird 70-77-0-0-70 bat. Won a World Series with the guy, in fact. He was a starter for me for 5 seasons (topping out a $1.8M) and averaged something like .206-.270-.340. He ran OPS #'s of .633, .687, .599, .591, 542. Do the Olympic thing and throw out the high and the low and you get a .600 SS who hit 17 HR's a year and had 42 + plays and 2 - plays in those 5 seasons. Lack of decent Splits didn't kill him because he was a guy who could do other things and his cheap rate allowed me to buy other sparkly baubles. But Splits remain the most important factor in reliably being able to predict BA performance.
Contact has a predictable linear impact on BA as long as the number is in the mid-range, let's say 40 to 70 or 75. A guy with a 70 Contact will have a predictable BA advantage over someone with a number of 50. But the rate of advantage declines after 70 or 75. A guy who has Contact of 90 doesn't get the same advantage over a 70 guy that a 70 guy has over a 50 guy. That makes sense in some way; Going from 50-70 involves a 40% increase, but from 70-90 is only a 28.5% increase. Anyway, when I look at guys with Contact near 90, or above, I assume they will perform no better than a guy with 75 Contact unless they are a lefty bat with world class speed.
I have become convinced that a low Eye negatively impacts BA. Time and again, I find guys with Eye of 40-ish having a lower BA than I would predict, often significantly lower. I don't quite find the converse to be true, that a really big Eye (90+) enhances BA, btw. But if an Eye has a predictable linear impact on OBP (and it does), a really big Eye seems to result in an increase above the linear projection.
BTW, RHB's will run a BA about 10 pts less than LHB's, all other things being equal.
And to Cub's point above, the one about vR and vL split correlation coefficients, if you combine the two the correlation will be much higher. So a 70 vL will have a .36 correlation and a 70 vR will have a .45, but a 140 Split total will be a much more accurate predictor.