I guess it depends on how deep the clay is, I don't like mudding, too much
cleanup, but there are two thoughts on the tire selection and why you would
want that tire.
First is the one you said, skinny and get through the muck to
the ground that is solid, first problem I see with that is chewing through the
surface to get to the hard ground, and the power/time that takes as well as
the momentum you loose, like being in a mud bog situation. I take the sand
rail approach, wider is better, float over the soft mud and kick tons of the stuff
to propel you over it, and being we have fairly light trucks, that is easier than
you think, although you aren't going to get there with 235 series tires, no way.
Mud and especially clay is sticky and slippery, there isn't an A/T mud/snow rated
tire out there that will do anything in mud, in my opinion the bare minimum tread
pattern is an M/T tire, BFG MTs or any really open tread patterns, I look at a BFG MT
as an All Terrain tire, they work well enough but still pack with the sticky mud.
I haven't seen many tires that shed the sticky mud like the Swampers do, I love them
and they work well for what I do, the mud racers use the Boggers to propel them
through the mud, and look at the sizes, wide profiles to float and aggressive to throw
the mud and not pack up.
Your problem is going to be, getting a big tire stuffed under your truck without major
mods and lift, and the power to turn the big meats, you get the tire and the power
and you will be amazed at the stuff your truck will blast through.
Food for thought
Wild