Yes, they'll easily escape from that container, and it will get worse as they get older. By the time they're four weeks old, they need at least 3/4 square foot floor space each.
I had a 16-18 inchs high cardboard divider in the spare bathroom where the chicks are. It lasted until they were old enough to get over it. Then, eventually, they became too heavy for it and flattened it out. I took it out as useless and in the way, so now they have the entire bathroom as their "brooder room", I'll just have to clean it later when they're in the coop in a couple of weeks.
I cut a large center hole in the rubbermaid container lid and then put hardware cloth or chicken wire over the hole you can bolt it down with washers and bolts etc. Then you can put the lid on and they cant fly out. I also feel the chicks are safer that way in case the cat gets ideas or the rope from the heat lamp breaks.
I didn't think about cutting a big hole in the lid, that's a great idea. I just draped some screen material and clamped it down with some potato chip bag clips. It works but I like your idea better.
I have my babies in a BIG parrot cage. I have no worries of them escaping or my cats getting hungry. The temp stays perfect and I can easily enough get them out and love on them.