It's kinda weird. Out of all Oscar's sidekicks, the one that's also performed by Caroll Spinney (Bruno) seems to be the one seen without Oscar the least.
The song I mentionend is the only time I can remember Bruno being seen without Oscar, though there are a few instances where Bruno and Oscar are in the same scene but Bruno doesn't carry Oscar in a trash can. When the gang leaves Sesame Street in search of Big Bird in Follow That Bird, Bruno is among the crowd watching them leave, and he's standing right behind Oscar's car. It's so weird that with all the wide shots of the street, Bruno isn't there much (though in the scene where Luis sees the boy playing with the map, I think I saw Bruno briefly appear, but it's hard to tell.. stupid window needed to be washed...).
And in The Muppets: A Celebration of 30 Years, Bruno is there, but doesn't carry Oscar (but they are both at the event, so I'm not sure if that counts as being "together" or not). Though there aren't many shots of Bruno there. At the very beginning he can be seen when the camera scrolls from the audience to the stage, and Bruno is shown, aparantly at a table by himself, right next to all the old people (and I think that's the only non-close-up shot of where the old people sat at). It looks to me like Bruno is the only character sitting at a table by himself, and it's weird that he's sitting so far away from the other full-body characters. But after that opening scene, there are a few shots where he's seen right next to the screen, a bit hard to see (I wonder if he was piurposely out of shot most of the time) until the finale, when he can clearly be seen in a close-up shot of the Sesame Street cast. And I think that opening scene is one of the few times when Bruno and Big Bird are in the same take as each other.
The other time that I can think of where Bruno is not carrying Oscar is an episode where Oscar was in his trash can min it's usual spot, and Bruno was carrying a trash can with Irvine inside of it.
Of course, besides the illustration of Bruno in Carol Spinney's book (and a handful of fan drawings, and some of my own drawings), the few illustrations of Bruno that I have seen didn't depict him carrying Oscar's can, though Oscar was in those scenes.