I am always looking for mesquite logs! I have a sawmill that I bought years ago mainly just to be able to saw my own mesquite lumber for my own hobby use. To be worth messing with, it needs to be at least 4' long and 10" diameter. Really, if it is short, then it needs to be bigger to make up for the shortness. Ideally, I 14-16" diameter log that is 6' long is perfect with mesquite. It does not have to be completely straight if it is big enough diameter since I will basically be making a square out of it, then cutting it into lumber.
Like I said, if you have something that is worth the effort to come down there to get, I would be more than happy to share the lumber with you. I do not sell lumber and do not sell stuff I make with the lumber. It is simply a hobby for me and stress reliever!
Oh, and I absolutely LOVE mesquite wood! Would never have guessed with my handle, huh?