"...higher SAP value will produce a soap with higher pH..."
Nope, it turns out this is not correct. Stearic and oleic acids (olive oil, lard, palm, tallow) have the highest pH and pKa values of the fatty acids commonly found in soap. The pH and pKa are lower for the shorter chain saturated fatty acids (myristic, lauric from coconut oil, PKO) and for the longer chain unsaturated fatty acids (linoleic, linolenic from sunflower, safflower, etc.). Source: Scientific Soapmaking, Kevin Dunn.
"...Should [high sap value] not cause more free OH- to be available? ..."
It sounds like you may be confusing the chemistry of saponification with the chemistry of ionic dissociation.
The sap value (grams of lye per grams of fat) is a measure of how much lye is required to exactly react with certain fat molecules
on a weight basis. The sap value is based on weight for our convenience, since it's hard for us to count molecules, but you really have to look at the molecules when talking about the chemistry of saponification.
If the fat molecules are smaller, more of them will fit into each gram (or per ounce) than large fat molecules. That means the sap value for short-chain fats (those with lots of myristic and lauric acid) will be higher than the sap value for long-chain fats.
Looking at saponification
on a molecular basis, 3 molecules of lye are required to saponify 1 molecule of fat, regardless of the length of the fatty acids that make up the fat. Out of that reaction you get 3 molecules of soap and 1 molecule of glycerin. That is the basis on which the sap value is determined -- that the lye must exactly equal that needed to saponify the fat. There is no OH- left over from this reaction to affect the pH.
It's like setting a table for a holiday meal -- you count the number of people to feed and set the table accordingly. It doesn't matter how big, tall, short, or small each person is. Every person requires a place setting, regardless of their size.
If you deliberately add more lye than the amount needed for saponification, then, yes, the pH would be higher. But that's a whole 'nother story.
edit: In the table below, "Laurie" should be "Lauric". Oops.