If you have the clearance to the bottom of the door, consider pan head screws. they give more bite for the screwdriver. But beware of door sag or you will scrape the paint off.
If you want to countersink and use flat heads, it can be done. You need to drill the holes first. Next, counter sink the sills a bit more than the head of the screw would need. The screw should drop into the countersink if installed without the sill cover. Now, get a piece of metal and make the exact same size countersink in it. Thread the hole. place the sill cover over this metal piece and put a screw in. Tighten the screw until the sill plate bends down into the countersink in the metal piece. You now have a sill plate that will accept a flat head screw, and a sill that will accept a cover with dimples in it.
I hope that made sense. It's a bit of work but I feel it's a nice touch that a nosy engineer, picky car show judge or future buyer will appreciate.