I had to use a large under bed tub for my big blood when her cage was not shipped on time. I used small clamps I got at Lowes. They were in the hand tool area and about $2-3 each. I had four and placed one at each corner. It worked for me.

You can also drill a hole in each corner and even in them middle it needed and use a bolt and wing nut. This will be the most secure but all the slowest to get in and out of.

Then like you already mentioned the luggage straps seem to be a good option. If your really concerned I would put two going and around the long side and another in the middle of the short side. That should keep the lid in place for sure.

Stacking things on top is a terrible idea. Tub lids are not very strong and then once you drill holes through it you make it even weaker.