FWIW, you've already given the answer to the question you didn't ask, specifically, "Will this work?" And the answer you gave is no, which is right.

I do understand the desire to cobble something together with no money invested in it just to see how low-tech and scrounged-up an item you can keep aloft. But there are two problems: First, this is an aircraft. And aircraft by nature are temperamental machines that simply won't work if they aren't exactly right. Second, you can't avoid spending money. Even if you could jerry rig up your current equipment, you'd still need a battery and power system and some structural materials.

So here's the honest to good right answer for you. If it really is about the building, skip the half thought out foam plate idea and get some plans for rubber free flight planes. These can be fairly complex and fancy if you want them to be, and a whole world of challenges to get them flying just right awaits. Your only expenses will be wood and covering tissue and a few props initially. If it's really about the building, get a Hobby Zone Super Cub, Mini Super Cub, or Champ depending on your budget. They are decent airplanes on calm days and proven fliers.