What about a way to escalate to my own team via chat? That way you field general inquiries and make it look alive (chat that goes to "leave a message" just sucks), but the moment someone is getting serious or has an in-depth question, you can tx them to me.

I think this is s good idea. At my last company, we did live chat, and it was very popular. But the constant interruption to our few developers wasn't feasible. And later, it was hard to get salesguys to make sure they were really always available. Taking the front load off is excellent.