The best stone you can buy is Cordierite Kiln Shelves, they come in various dimensions. You can buy them at pottery hobby stores, where they sell kilns and other accessories.
The one in the picture below cost $18 with tax, is 5/8 thick 15"x16", it can handle heat up to 2200 F.
As for care, do not wash, or season, if it needs cleaning use a damp clean
cloth and NO soap, if stuff is burned on, scrap or brush off, just like your bbq grill. If scraping doesn't work get your sander out and sand it down to the stone. When I bought this stone from a local store, the person helping me told me (after I told him what I was going to use it for) that many pizza restaurants buy these for replacement stones in their pizza ovens.
Your plan is to use in the weber, unfortunately the hinged grate idea isn't practical. You need lots of briquettes to get these babies up to heat, around 600 to 800 deg. You need to remove the stone and the grill to make this work properly.
I outlined and crossref a previous post about the procedures here
One possible way around removing the grate, is to make a large funnel, that would fit perfect into the half round grate hinged opening. You would then need to have a heavy bend rod, to stick under the grate to spread the briquettes evenly. However read in my post where I found the best layout for the briquettes was ring on the outer edge.