TubeNet

That would be Ron Caswell. If you search Tubenet for tips on playing like an elephant, I bet I know who you're going to get, and it isn't snuffleufigus.

I'm neither Ron nor an elephant, so I'm just going to guess that it might help to experiment with a smaller mouthpiece. Not that your mouthpiece is too large to play beastly sounds (I don't even know what it is), it's just a trick to get a fresh start. I'm also going to guess that it will be fine to just play tuba and the kids will get it, so don't go overboard on this.

I always find a much bigger mouthpiece makes me sound more elephantide until I get used to them - then they just make me sound worse!