Of course he'll push. Tsukikage wants to buy muffins for the kids anyway, so naturally he'll want Speedwagon to also eat. Tsukikage doesn't take any himself, of course, instead just looking down at his hands and thinking of a topic.

...

"So how did you find out about my shop?"

Tsukikage internally kicks himself for asking a question he already knows but it's not the worst question. After all, 'Charlotte' knew but 'Tsukikage' didn't. He'd ask for questions more about Speedwagon in a second.