I can't speak for Kevin or Lynn, but there probably is a good reason why they are both using a single LC network in the power supply. Perhaps Kevin could tell us?
You undoubtedly have noticed that there is a heated debate going on on the tube diy forum on this topic, so please take anything I say as my personal opinion. If you want to be on the safe side, stick to Lynn's schematic (this is what I would do anyway).
About the capacitors, I think it is benifical to keep the first cap low (as much as output voltage allows) because a higher value will also give higher pulse currents on the rectifiers. Use a good high voltage wideband snubber type here. The last cap should be able to deliver the currents requested by the output stage when it transitions into class AB, so it should not be too small. If you are going to experiment with exotic capacitor types I'd start with this WE oil type that is closest to the output stage. This one is probably the most audible.
I don't think it's easy to design a good LCLC power supply. In fact, I think that power supply design is often underrated/underestimated. I certainly don't think I'm qualified to design one, so I'd build the simple one LC section as per Lynn's schematic first. If you breadbord it, you can always experiment and see if you like it with two sections later on.
A nice and easy experiment to do, is to add a RC section. This lowers the ripple quite effectivly and is less likely to mess things up. Another interesting option I still want to try is to add a shunt regulator on the output stage.