I know it's tight back there and access is limited but can you clamp a small pair of vise grips to one of those "wings" on that basin nut and pull it snug? You might have to go one at a time till its snug. I was thinking like 4 or 6 inch vise grips. Like someone else said who cares if you booger the finish on the nut a little it's hidden from view.
It's worth a shot, if you can get them in there, but I'm pretty sure that the nut is chrome plated plastic, so go VERY EASY on this route or you may rip a wing right off and be in a worse position.
I'm pretty sure the Basin Buddy is made to be used in place of a basin wrench, on the valve clamps, which are normally this inaccessible, not on the drain, which is only this inaccessible in the case of pedestal sinks like this.
I also think it wouldn't line up with the splines on this nut, but I do like this way of thinking.
Superbec makes a good point. Some silicone o-ring grease on the rubber may make it possible to get this joint sufficiently leak tight when just finger tight.