I just checked Kreg's web site and they have Stainless Steel screws With "Maxi-Loc" heads, whatever that is, for about $22 a hundred. They did not show a clear picture of the maxi-loc head but it resembles their standard screw with what appears to be a washer head. Various lengths. Good luck.
I can't recall the name brand or why I needed some, but my local 'lumberyard' carries these.
I have a box (in my vehicle) of #7 1.25" long washer head screws, purchased with in the last year. These are the 'yellow' coated exterior screws made for use in pressure treated lumber. If I had to guess I would say 'Grip-Tite' was the name brand. The lumberyard is locally owned, and sources from a variety of suppliers, including the 'Do-It Best" chain; I just checked their website, but it does not show all the products they have on their shelf, so I can't link.
So I don't know if this is the size you need, but it confirms the product availability at least.
If you can't locate what you need, I will pursue the details for you.
I recently put a plastic roof on an outbuilding. The screws that were needed to attach the roof panels (24" x 10') to the purlings were exterior screws with a washer. They were purchased from HD and found in the screw/nail aisle. The same screws w/ washer were used a few years earlier when I put a metal roof on another outbuilding. Note: the screws have a hex head and will stick out. I'm guessing they will not be attractive for a screen application, but would work.
They have held up for a few years now in a similar application. My screens are permanent, but I have 1/4" thick plexiglas panels I put over them for the winter. The toggles are 10 mm exterior sipo (African mahogany) Festool Domino tenons, and the spacers are aluminum tubing I cut to length on my radial arm saw *. The toggles hold the panels against triple bead mahogany 3/4" x 1/4" strips that pin the screen down. The toggles and screws stay attached to the porch year round. The pic shows a panel still mounted to the screen and one already removed from the door. They will all be taken down and stored in the next couple weeks. No tools are needed to twist the toggles a quarter turn during installation and removal.
* To cut aluminum tubing safely and conveniently on the RAS, I drill a hole in a piece of scrap to fit the tubing, clamp it to the RAS fence, push the tubing in and cut it. The fixture keeps the offcut spacers from flying all over the shop.
This is the cowards' way out but if you enter the search term washer-head exterior screws into Google a number of hits show up and many of the topmost show SPAX brand screws at Home Depot. A starting point anyway.
Because it was so simple and cheap, I'm afraid I tossed it when I was done making the spacers. S, I will have to try to explain it in words. Start with a block of wood that you will be able to clamp to the saw fence. Raise the blade so it doesn't cut all the way through the block but does cut through the tube. Drill a hole in the block the long way, that is the same diameter as or a smidgen bigger than the OD of the aluminum tube, parallel to the fence and the table. Lay the block against the fence under the blade, and clamp it on the left end so that a half inch of block remains to the right of the kerf, assuming you want half inch high spacers. Cut the kerf.
Shove the tube in through the left hand end of the drilled hole, and stop when the other end of the tube is flush with the right hand end of the block. For consistency, it helps to use a stick fed in from the right hand end of the fence and up against the block, to consistently stop the tube at the correct distance. Cut off one spacer. Push the spacer out the right end of the block with the tube. Watch that the spacer doesn't roll off the table, but don't stick your hand under the spinning blade. Rinse and repeat.
Note that the air being blown forward and down off the blade can also blow the very light cut spacers off the table. You might want to build a catching mechanism, made for example from some PVC pipe and a cup.
If this isn't clear enough, please get back to me. If I can find any aluminum tubing, I can easily make another block and take a couple of pics.