Leaving the lids on will work if that's the route you decide to go, on the racks I've built I usually will have a gap of about 1/8-3/16 of an inch, but they are designed for adult BP's.
That being said, I don't see a baby BP making it through a 1/8" gap, but the gaps on yours appear to be closer to 1/4". But it also appears as though the shelf is curved kinda upward, so try putting all your hides and bowls in the middle tub and see if it's gap opens a little and the lowest shelf gap closes.
If that plywood has some flex to it though I'd definitely consider using lids on the bottom and top tubs, but make sure both the lid and sides are ventilated well so there is enough airflow.
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