The bat wing style reflectors from HTG supply will fit into your footprint. They are an American company. The 400 watt bat wing style reflector they offer is 14" long x 13" wide . Taking into consideration the height of your room and the angle of light deflection from the hood itself, this reflector would be good for you. A little off topic but let me get to you before you start building out. Before you do anything, make sure you have a great ventilation set up. Because of the small cubic foot volume of your room , a 400 will really heat things up in there (I'm going off topic but stay with me just in case you didn't consider this already) so just remember that the smaller a room gets the more ventilation it will need for the same amount of heat sources it has.
You could also buy a pack of pebbled finish 95% reflective German aluminum from Sunlight Supply. It costs about 150 USD + shipping and it is quite easy to DIY 4 bat wing style reflectors with this sheet aluminum, with a little help from a DIY sheet metal bender made from 2-2x2 handi panels of plywood and a half length of piano hinge (and a few sheet metal screws and drywall screws ; also a jig saw and cordless drill , don't forget the stainless steel music wire and 16 gauge barrel crimp sleeves). For the size of reflector you need, you would probably be able to make 8 of theses fixtures with one pack of that sheet metal.