I would like to be able to use the rectangle tool to draw a map using these large tiles, but it still moves by 1 base size instead of 2. Is it possible to use the large tiles to draw a map?
No. If all of your tiles are large, then increase the tile size in your tilemap.
Your big tile just covers some other tiles around it, but it doesn’t fill them. To the engine (or tilemap), there is one tile assigned and the tiles around it are empty, even though there is an image there.
You can adjust texture origin of your big tiles, so that the tile maybe isn’t at the center of the image, but top left or something. Might be helpful. That way your big tile covers for example one tile right and one tile down instead half a tile on all sides.