BCX shouldn't warp on you, especially at 5/8" thick. It's rated for outdoor usage*, so as long as you treat/finish it correctly, you shouldn't have a problem with it warping. You could probably get away with 1/2", depending on how much weight you actually add with the mosaic.
As far as finishing/treating it goes, that depends on how you are using it. Gluing things to one side will usually seal that side, depending on the glue you use. Using a standard wood glue might not stick to your mosaic pieces and may not seal that side, but using a silicone or epoxy glue will seal it, while also likely sticking to your mosaic pieces.
Simply painting the wood should be enough to seal the edges and back side. You can also stain and use other finishes, like shellac or varnish, if you want a different look to the back. If your artwork is going to be outside, make sure the finish is rated for the normal moisture and temperature range in your area. Example considerations: don't use an interior paint for an outdoor exhibit, don't use a finish that isn't water resistant if it's going to see a lot of precipitation or humidity, don't leave bare wood in a low humidity and high heat desert, and don't bother with an expensive or time consuming finish for an indoor exhibit unless you really want to.
You can get chemically treated plywood for outdoor use, but paint and glues might not stick to it. And if it's chemically treated, you don't want to paint or seal all edges.
*Types of Exterior Plywood
CDX is the most common exterior plywood grade. It comes in 3/4-, 5/8- and 1/2-inch thicknesses. Appearance stamps on plywood range from cabinet-grade (A) to construction grade (C and D). Exterior plywood carries an X stamp, which identifies it as suitable for use outdoors. Beside CDX, you can also find better grades, such as ABX, ACX and even BCX.
You can also think about using a sheet of aluminum or steel, although that'll likely cost you more than the plywood. I'd suggest 1/4" thick for rigidity purposes. You might be able to get away with 1/8" or 3/16" thick, which should be less expensive, but then you could get into more issues with expansion and contraction of the piece warping it, if it's going to see extreme temperature differences.
With steel, you'll still need to seal the unglued side, since it'll rust even indoors. That's as simple as painting it, most of the time. If you want a better look or a more durable finish, you can even powder coat it. Aluminum doesn't need a finish, but it can look better if it does. Or you can do a wide variety of finishing to the aluminum itself, such as brushing, scuff sanding, or polishing. Aluminum can also be anodized a variety of colors.
With both steel and aluminum, to reduce costs, you can use expanded sheets, but then you risk warping again, as well as the glue not having as much surface area to grip. And if you want something that gives more surface area, don't really care about costs, and has a really neat pattern, you can use diamond plate.
Even 1/4" steel and aluminum can be cut with a circular saw with an abrasive disc, an acetylene torch, or plasma cutter.
There are definitely reasons to use a sheet of plastic for your backer. It can make it light weight, transparent or translucent, weather resistant, chemical resistant, warp resistant, and a few other things, too. The problem with plastic backing is that it's harder to find glues that'll adhere to plastic and other materials, such as metals. It's not impossible, but they are usually more expensive and may not work as well.
Even though plastics won't warp from humidity, they can still warp, or even crack, due to temperature. Many plastics are stronger than glass, but they still tend to have properties similar to glass. A piece of polycarbonate is much stronger than acrylic and many acrylics are much stronger than glass when it comes to an impact, but due to age and high heat, can turn yellow or become brittle. There's outdoor rated plastics for signs, but they cost more, too. Outdoor rated plastics may also be harder to find and when you do find them, you might only be able to buy them in 4'x8' sheets.
Plastics likely won't need a finish, unless you want to protect them from UV or reduce glare, and even then, you can sometimes get it from the manufacturer with that coating. Outdoor plastics usually have UV coatings on them already. Plastics with a P95 or P99 coating have an anti-glare coating on one or both sides. Frosting is also an anti-glare treatment and has a completely different look, like frost on the grass in your lawn on a chilly winter morning. The other (P95 and P99) coatings are clear and don't interfere with color or translucency.
For the size you're looking at, 1/4" thickness might be all you need, but 5/16" to 1/2" might be better to add rigidity. Plastics tend to cost more linearly as you increase the size and thickness. A 1/2" piece of acrylic will usually cost twice as much as a 1/4" piece the same dimensions. However, 1/4" plastics are much more common to find and can come in a wide variety of colors, while most of the 1/2" plastics I've found come in clear, white, and black, with few exceptions, unless you want to spend some serious amount of money (over $500 for a 4'x8' sheet) to get it.
Believe it or not, plastics can be cut like glass or wood. You can score plastics and they will snap on the score line like glass. (And just like glass, if you don't do it right, it won't break on the score line.) They can also be cut using a finish blade on a circular saw. Some plastics, not all, can be cut using a laser cutter, although you need a pretty large area laser cutter for the size you're looking at. Many plastic suppliers will do cut to size for you, too.