I would shift the lamp over to one end or the other, rather than the middle, so there's a more clearly defined cool area for them to retreat to if needed.
You can also pick up pretty cheap a pair of little thermometers to place on the ground. Technically they say you don't need it bc you can just observe if they are huddled under the lamp, vs. trying to escape via an edge, or just happily milling about. But it may give you the peace of mind while you're sleeping that the surface temp under the lamp isn't TOO hot, and I'd watch long enough that you're satisfied they are comfortable, and the temp is stable before going to bed.
The tub / gate lid is probably fine for now, but they may soon outgrow it. Just try to be responsive as they get bigger that they aren't over crowded, but with just four, it might not be too bad. Their presence isn't going to magically attract mice or rats, but open food might, so just keep their feed and treats in a sealed tub or bucket if you are wary about pests.
I was also going to ask if you have a way to secure the baby gate? That way you can also mount the lamp above that and prevent the lamp from falling directly into the pine shavings of it were to come loose. We have always brooded in a metal dog crate, and then wrapped it in cardboard to keep the chicks and pine shavings in.