How about cutting off the bottom then gluing on wider pieces of styrene.  Then shape the edges to show the bulges of a flat.  Painted and stuck in the ground and weeds should be convincing enough.

I have used a sealing iron. It has a teflon coating so the plastic does not stick to iron.  It takes some experimenting to get the right temperature setting.  But when you do, you press the wheel down against the hot iron till it squishes the desired amount.

Many hobby shops carry these for use applying the stick-on covering material on RC planes.  I bought mine mail order from Tower Hobbies while I was still into RC, but found, with the right setting, that it was great for melting and softening styrene.

I was worried about getting the rim part too hot and having it change shape on him.

  Cut the lower piece of tire off leave the rim intact of course, then with some model clay sculpt the desired shape of a flat tire. You can practice sculpting before making ant cuts. Just my 2cents.

    I was gonna say let the air out of them but I'm not sure the humor would be seen.

   Good luck, given what you have accomplished before I'm sure this will be a snap.

Hmm'

    This is the Wheel/Tire type that is two piece plastic. Right? Bets thing you can do is try to find solid tyres and wheels and then Put the thing in Boiling water. Use safe procedures . Then press it down on a piece of glass . You get that flat tyre pooch and the rim will be unaffected if you don't press to hard.

I usually just throw a sanding drum onto my Dremel, and sand the bottom of the tire flat. The sanding drum will heat up the tire a little and I just "squish" the now exposed edge's of the tire down.

Thank you all. The hot iron may work, will give it a try. Tanker, it is a one piece plastic wheel, both the tire and the center of the wheel are one piece. 

Here 'ya go.

6 pictures say it all.

I would cut the bottom off and then sculpt the bulges with some epoxy putty