Plywood is a lamination of several layers of wood set with the grain rotating with each layer. This actually helps resist warping, but does not eliminate it (as you can see). This happens because of some flexibility in the glue that allows a gradual creep.
The usual solution is as you suspect - laminate the plywood to something else. This could be a metal bar, either aluminum or steel. Perhaps the most effective would be angle stock (metal forming an "L" in cross section). A flat edge of the angle is screwed to the back. The perpendicular edge then acts as a brace.
If you do this, you need to first flatten the piece as much as possible. I would recommend placing it face down on a flat surface, possibly with a sheet below it to protect the art surface. Then weigh down the corners with very heavy weights, such as full paint cans or barbell discs. Lay an angled bar across the full width near the top and another across the bottom. Place screws every 6 inches. The screw length should be long enough to go through the angle stock and 3/4 of the way through the plywood.
You can then rehang the piece using a wire threaded through holes at the ends of the angle stock. If you really need to use velcro, you could attach wooden filler strips to the upright edge of the angle stock, and attach the velcro to the back of these strips.
An alternative bracing can .be done with strips of plywood, al teast 3 inches wide and 3/4 inches thick. These can be glued and screwed to the flattened panel instead of the angle stock. If you use a strong wood glue, such as Titebond Ultimate, this will help prevent creep, as will the screws.
The advantage the angle stock method is that it is reversible (no glue).
You do not want to cover over the window with a TV. First of all there is the glare point that Paulster2 points out. That alone should make you take stock of this situation and see what a bad idea this is. However there are a number of additional considerations as well.
Windows have a role of safety exit possibility if you ever get caught trapped in a room during a fire. This for both outbound escape and for inbound access by emergency personnel.
There is a reason windows are added to a house. Obviously it is to allow light in so that the room does not look like a cave.
If you have no usable wall space to mount the TV then get your self a console table. These do not have to be that expensive and you can even get them with an integrated post assembly that permits mounting the TV bracket with tilt and swivel capability. Best with this approach is you can move it around if your change your TV room layout or go off to a different house.
Talking about moving. Say you did move after you had mounted a TV over a window. Think of the mess in the window trim that would need repairing before you could sell.
Lastly, putting a TV over a window is just weird.
Best Answer
Why don't you consider a free-standing TV stand, similar to this? This would easily handle a large LED TV up to at least 50", is easy to move and doesn't require covering your window.
Covering the entire window with plywood then mounting an LED TV on it, and having it look good, will take some doing. I would think that you'd want to screw 3/4" plywood into the window frame (the studs/cripple studs, ledger and header under the drywall), sand it well, put some kind of attractive edging or molding around it and paint it. If you're going to hang a TV of any size on it, I wouldn't go thinner than 3/4". Or if you do go thinner (like 1/2"), I'd probably put a cripple stud centered in the window to brace it. To secure that, I'd put a block on the bottom and top sill, screwed into the window frame, then I'd screw the cripple stud sideways into the blocks. Then I'd screw the plywood to the cripple stud as well as all around the window frame.
Use screws for everything you attach, don't use nails anywhere.
If you do all that, then you should have a solid base, and you would just use a standard LED TV mounting bracket screwed straight into the plywood with substantial screws (here again, using 3/4" instead of 1/2" plywood helps you).
Something along these lines:
I presume you're doing this to mount a larger LED TV. Most of the articulating-arm style of mounts are for smaller TV's and monitors, although there are some that are designed for larger panels, similar to the pictures below. Of the two, I'd probably prefer the first example. But do note that the second example has a deceptively large mounting plate, with holes in a much bigger pattern than the standard 75mm and 100mm pattern you see on smaller mounts.
Also keep in mind, though, that with these types of mounts, all the weight of the panel is concentrated on a much smaller mounting surface and it has to be mounted very securely to a strong surface. The top screw(s) in the base, in particular, will be bearing the lion's share of the weight, and if you pull the TV out from the wall the force on that top screw(s) will be magnified. So if you do this, center the mount directly on the cripple stud that you positioned inside the window frame, and using nice long screws (2.5" to 3") to secure the mount.