just remember, the breakers only job is to keep the wire from over amping and melting, so breaker size is determined by wire size and type. Also, I can't tell what size the wire leaving the panel is But hopefully its #10 since its hooked up on 30 amp breaker. The problem is gonna be the wire size on down the line, so you're right about possibly reducing breaker size in sub panel.

I would also see what size the wire is coming from breaker to that sub panel and make sure breaker feeding sub panel is the right size (not exceeding wire amperage rating). Just a guess, but it looks about right size and type, due to the fact that its stranded ( will handle more amps). I would hook it up to a single pole breaker (from feeding panel) as well. I don't know if code requires that or not, but like i said. I like for my electrical to be perfect. But its hard to say without looking at the whole thing.