Foam sandwich construction as listed above will definitely safe you weight. my 5x10 home built tear is about 900 lbs camping. I do need to weigh it to confirm, but the shell was 300 lbs before I put it on the trailer frame. foam sandwich is strong and light, see my avatar photo for an example of what 1 inch of foam and 2 sheets of 3mm ply can do. I am 180lbs and I am standing on a 14 inch cut out for the roof vent.
In addition to sandwich foam, you need to really ask yourself how stout/strong certain areas really need to be. I was kind of obsessive about saving weight. why use a 1x2 when a 1x1 would work? why use 1/4inch when 1/8 is sufficient?
another area that you can save weight is the trailer frame. you don't need a lot of heavy steel to have a strong, safe platform for your tear. you are not building a house where you need a true foundation. they box of the trailer shell should add strength to the frame, not rely on the frame for strength. my frame is a simple double square with diagonal braces for the tongue. only 3 cross members in total, that includes the front and back. There are nearly 5 ft unsupported areas where the cabin/shell floor has no steel under it. It saves saves weight when you cut out the unnecessary steel and downsize what you have to thin gauge tube vs thick angle. additionally, go with a smaller axle and tires. I'm not saying the little 8 inch wheels, but you do not NEED 15inch plus wheels and a big heavy 3500lb axle. all of that adds weight quickly. my tires and axle alone weigh in at 150 lbs, by far the biggest area where I could have cut weight. smaller 12 inch wheels and a torsion axle would have weighed half that and worked just as well for a tear.
ounces add to pounds. don't miss an opportunity to cut a pound or two, because if you say "its only 10 lbs, that wont make a difference" and you do that 10 times you just added 100 lbs to the finished product.