Does he get free play during the gym class or does he have to follow instructions? Do you have indoor play spaces where he can climb? We're so lucky here that we can really go out all to parks all through the year. I'd have struggled with my very active boy if we had to stay indoors.
Does your DS like water play? DS played in the bath a bit if we didn't go out... Also he would play happily with a couple of utensils, some water and some washing up liquid for bubbles.
What about finger painting?
Why can't he climb the things he's climbing? Is there a house rule that you don't climb the arms of couches, etc.? Is it unsafe because he'd be too high and could fall onto something hard, potentially doing serious physical damage? Is he getting into trouble from you or because he's in danger and can't extricate himself? Is there anything he can climb inside the house?
What about indoor rock climbing? He's pretty little but I've seen some really tiny kids get into that.
Is he climbing for the sake of climbing or is he trying to reach things? If so, are they things he can have? If yes, maybe put them down lower and if no, maybe 'out of sight' would be better than 'up high'.
Is he climbing for attention?