How can I strengthen my melted bead art?
I was looking into the type of plastic these are (LPTE?) and found a comparison of different fuse beads:

http://houseofgeekiness.blogspot.com/p/not-all-beads-are-created-equal-when-it.html

This guide casually mentions one technique, also mentioned by Matt in comments: fuse both sides. 

The other major factor here is trying out a different brand. The guide recommends Perler, or generally away from the overly cheap beads. Not all plastic is created equally.

If you can try out other brands, that's what I would test first. It's something artists tend to do with supplies, anyway. Different brands give different feels or effects, and some just stink. 

Another tactic is to make sure you're melting consistently. If you use a plastic ironing sheet, instead of a paper one, you can eyeball the level of melt in any given section of the piece.

If you don't need both sides of the image to be on display, then I would simply adhere something stiff to the back. It might be possible to use thin, clear plastic sheets, if you can find an adhesive that works and dries clear / without streaks. If so, that would let you add external strength while still viewing both sides.