@twittwootwittwoo
Please, please don't feel shit. There is absolutely nothing wrong with a bit of Cbeebies in amongst everything else you've been doing! It's all about balance. My two DC have played all morning, but they're watching a bit of TV with a snack at the moment. DD(6) is isolating from school until Monday, and so far today they've done some Lego, play food, cars, chasing each other around the living room... and watched some TV.
We usually play or go out in the morning, watch a bit of TV with a snack, play some more, get DD from school, watch a bit of TV with another snack, play, dinner, Cbeebies Bedtime Hour, stories, bed. It would be different if they were plonked in front of the TV all day, but they aren't. It's mixed in with other things.
Please don't worry about a bit of TV.
And OP, there's nothing wrong with what you're doing, either. When your DC is older I wouldn't have it on all day, but at 2 months? Do what you want!