"Yeah, let's let Sinnosuke go" is how it feels to me (which is close to your second, revised version). I still feel like it's permission more than coercion. That may have to do with the に instead of を (i.e., しんのすけを行かせよう) as well as it just feeling like this is more about letting him go on his own than making him do it.
そうだ can be translated a few different ways, such as "that's right" or "yeah". だ isn't an abbreviation of です, though, any more than 行く is an abbreviation of 行きます. They're two different forms, one familiar and informal, and one polite and formal.