Haha yeah seems in my experience the people who want jobs done such as the one you as asking about always wait until the ultimate LAST moment... seems like an afterthought to me. I don't know about DIRECT to HAT printing, but I know that if you use Roland's Heat Soft material for this job and a Hat Press you can get away with it rather inexpensively

it's this material
http://www.rolanddga.com/products/media/detail.aspx?id=152 just use your in-house printer/cutter for the job and be aware that any hat with a center stitch (2 panels not sure the actual name of these types of hats) usually don't look so good because the heat transfer material might not get into the crevasse in the center area.
Personally when people come to me with such a short turnaround I give them options. Option 1 with LONGER turnaround is usually almost 50% cheaper than last minute RUSH orders, after I tack on a "rush" order to their second option (short turnaround) then they re-consider their turnaround OR just fork up the monies. I am sure other people might have better solutions

these are just my 2 cents.