The product is a month behind. Do you go ahead with the launch and launch with a product that isn't fully GA? Why or why not?

It is a judgement call based on opportunity cost and organizational culture. 

I recall Salesforce launched its Einstein Bots at Dreamforce, 6months to a year before the product was really available. As a competitor (Helpshift) this put us on the back foot since they were a dominant player in the space and their 'launch' created enough of a buzz in the market. Many large B2B software companies tend to adopt this approach to gauge market interest, fill up their beta customer pipelines, etc. There is relatively little cost for them to pursue this approach. 

If you are a smaller company, with fewer customers and resources, a similarly sized launch could stop any other GTM launch/activity from happening and could happen at the expense of lead gen or even confusing account teams who aren't used to selling vision first. Sales teams will then fill in the gaps and you could have new customers sold on a product without a clear feature list. Eventually leading to churn and technical debt.