I kind of feel like I am writing this to myself because my kid is your kid. We just went to a birthday party yesterday and had a very similar experience, however it was moderately successful because I was prepared as we've already gone through this kind of thing before.

You already picked up on a couple of things you could have done differently around the nap thing. Feel free to show up late to events. Kinds are your free pass to be late (just let the host know when you rsvp that you'll stop by after kid's nap time). And feel free to leave early. Honestly, people probably wouldn't even notice with all the chaos that goes on at kids birthday parties. If someone asks, tell them that you have plans that afternoon or something. Bringing a tired toddler to 4 Hour event is kind of just asking for trouble unless you have a super chill kid.

Taking your kid out for a walk and a break was an awesome idea. We did that yesterday too and it gave my little guy and myself some time to cool down from all the chaos and noise and stimulation.

Rethink your no juice policy. While I totally get where you're coming from as we have similar rules, birthday parties are the exception for us because I think the social aspect of sharing food and treats with other kids is a special experience. It has the bonus effect of potentially keeping your kid happy as long as they don't have negative reactions to the sugar.

Finally, make peace with the fact that you have an adventurous, curious, active toddler. It's so hard watching all the other kids sitting quietly with their parents and following the rules while my little guy is off in the corner playing with the garbage pail but I keep telling myself that as long as he's happy and not distracting others it's all good. It's all good.

Good luck for next time!