What is the minimum clearance I need between the truck cab and camper overhang?

I was assuming the 2x4's were standing on the narrow part so they could possible topple sideways. If they are going to lay flat I would say 1/4 inch ply is more than enough.  If you want the 1/2 inch extra lift you could box the 2x4's top and bottom with the 1/4 inch ply to make it more rigid when adding or removing .

You can also probably go with 2x3's and 1 top coat of 1/2 inch ply if they are going to be laying flat. I went with the 1/2 inch in this suggestion in case the camper walls are thicker than the 2x3. In that case the extra thickness of the 1/2 inch would better spread the load.

Depending on wall thickness can you get by with 2x2's instead?

I would go with the three frames of 2 x 4s and probably 1 inch ply on top. You already need a little more height and there is a big difference in strength between 1/2 inch and 1 inch ply.

I got my truck and thought I'd post a follow-up.  I set my camper on my bed supported by some 2X4s (raised 1.5").  That gave me 2" minimum clearance at the center of the cab and about 3" at the clearance lights at the front of the cab.  Just looking at it it is hard to imagine how I could get enough flex to make contact and even if I did make light contact at the center of the cab it seems the sheet metal would just flex and bounce back out.  However I only have an inch at the shark fin antenna at the front of the cab so I need to relocate that unless I want to raise the camper another couple of inches.  

Now I need to figure out how to make a platform.  Bigfoot campers have a perimeter frame so it doesn't do much good to have it supported other than around the perimeter of the camper and basement.  I could make a 2X4 frame to support it at the proper locations around the perimeter and basement but it would be awkward placing it in the bed and keeping it at the correct location.  I could attach it to the camper to make loading easier but I don't like the idea of putting a bunch of screws into the camper.  I could also use plywood on top of the frame to hold it together but then it would be quite heavy to load and unload.  I'm contemplating dividing it up into three separate platforms to make it easier to put in and out them into the truck, maybe using 2X4s and 1/2" plywood.  

Any comments from Bigfoot owners?

New campers have 48" between the floor base and underside of the cabover. This is a safe height to use as a reference against yours.

In addition to the static clearance and frame flex, don't forget to also consider the movement that your tie-down system will permit.  You might even want to test this and see what happens in a panic stop situation.  I'm thinking you could use a piece of something like the crunchy FloraCraft FloraFom from Michaels.  Tape a small piece to the top of your truck and drive around.  If the cabover touches the foam it should leave some sort of witness mark from which you can better evaluate your dynamic clearance.

So did your camper fit with enough clearance to the cab?

I built a 1" insulated lift for my TC thatworked fine for several years but it only provided 1/4" cab clearance. I recently built a 1-1/2" insulated lift which gives 3/4" lift. It has never caused an issue. I may have posted a spammy video here about it last year. The new lift also helps center and keep straight in the bed, how the TC sits.

I’d say not may have the “exact answer”. 
1.5” seems a bit light, but may work. Probably fine under normal conditions but hitting a big dip or frost heave at highway speeds might test that theory. I’d shoot for a bit more. Nothing a sheet of plywood or 1” foam under the bed mat won’t cure. 

congrats on the new rig!

Also check the distance above the rails. My 'foot dealer had to run and get a foam sheet to get me home when it appeared the camper would hit the rails. I now have a 2X6 built platform that keeps my camper centered also.

There is a lot more movement up there than many realize with truck frame flex and cab over bounce.  I'd want at least 3" clearance, which would be easy for you to do with a flat frame built with 2x4 or 2x6 lumber or a sheet of pink/blue insulation board for support