My best recomendation in that area is either chez l'Ami Jean (rue Malar, 7th) or la Régalade (av Jean Moulin, 14th). Both have 32 eur menus. It actually will exceed your 100$ mark, as are le Pré Verre and Constant.
Actually, the only real possibility of great food within your budget that I know of are la Cave de l'Os à Moelle (20eur menu, great wines, you have to cut the bread yourself and seat at a large common table -- it's in the 15th), and BE for lunch (It's the Ducasse bakery/shop/restaurant -- try their truffle pasta at lunch for 13eur).
I don't know if I'd call it regional, but it's excellent: try Le Pré Verre, www.lepreverre.com. Contemporary French cuisine with a slightly exotic touch and only 27.50 euros for three courses. There's also Café Constant on rue St Dominque, super chef Christian Constant's affordable cafe.