As someone who has gone through probably thousands of sheets of lauan in my 27 year scene shop owner career days, I can tell you that it will work if you do it right. I'm not sure that it will be any cheaper in the long run, if you value your time a bit, than using Gatorfoam, though.
Lauan is a broad species name that really encompasses a whole family of tropical mahogany tree types, and they vary tremendously in some respects, most notably the tannin content. The blonde stuff has very little, and the dark red has the most. It will migrate through any water based product over time and stain it from the back, and it is very acidic, so you will want to seal it. That needs to be done with a solvent based barrier coat. I found the shellac based BIN primer or KILZ to be the best at sealing it from leeching tannin. To prevent warpage, you will want to prime both sides. I would then give it a light sand and adhere the canvas to the wood with Miracle Muck/PVA adhesive, or one of the giclee coatings that claim to do double duty as an adhesive, like Glamour II.
A recessed wood frame on the back that is well built, or better yet, one with a second skin on the back, will hold it completely flat, with only a 1/4" canvas wrapped edge. The second skin is what engineers call a stress skin, and it gives it great flatness and stiffness with even a crappy frame. That's how hollow core doors are made, and they rarely warp, even with low quality materials.
When all of this is said and done, though, I question if this is a better way than just using Gatorfoam with some museum cleats on the back.