The screen on top of the joists is pretty much the standard method of dealing with this. Use fiberglass screen, not alum. Much easier to handle and not a problem in this low abuse environment. Pull it tight as you staple. Compressed air can help blow stuff out, but yes, it can accumulate. The other thing to think about here is that, since this is a screened porch, you will get a lot less debris on the deck boards that could fall through and get stuck there. Most of the coarse stuff like leaves, tree seeds, cottonwood/dandelion fuzz, etc won't get to the deck boards as you have a perimeter screen on the enclosure. So it comes down to dust and whatever bits of food and stuff people drop. You may have one really bad cleaning session at the end of construction where you need a day , a compressor/blowgun and a vacuum, but after that it gets a lot easier.
If you want to go underneath the joists, screen will look really bad down the road. As crud builds up there, it will look like a series of crud hammocks that you made. A better choice might be to make the underside solid. Slope it well to one end for drainage with tapered shims or something, and then tack up plywood to the bottom of the joists. You can use the space for running wires/lighting for below the porch too. Make sure the drainage openings are screened somehow too. Probably with a frame you can remove from the outside for cleaning. And if you are really thinking ahead, you can create access at the top of the slope so you can get up in there with a hose to clear out crud down the road....
Neither method is painless. The screen makes it much harder to lay the deck boards on an elevated deck as you can't poke your body up through the joists from below to work on decking since the screen is in place, so you have to step carefully on top with staging etc. The below joist method is much more expensive, and a lot more total effort.
Edit: One more point - you are building this in the high heat of summer. You probably will not see much more expansion than this, so perhaps 3/16" is too much given the current conditions? It will only shrink from here, unless you are in a real cool spell when it is installed. If so, then the screen on top is looking even more attractive. IMHO