Classification model for recommender system?
One option could be to model the output slightly different. Instead of returning the type of product 1, 2, ..., n or 0 in case of no product maybe you could return a list of pairs (product, probability), if the highest probability is the one corresponding to 0 i.e. "no product" you could just recommend the 2nd best. 

An advantage of this approach is that you could recommend more than one product, maybe the top 3, or a list of products whose probability is higher than a specific threshold, etc.

If the customer is new and there is no information you could base your recommendation on, maybe you could recommend the product that you would offer to a similar customer (if you can cluster the customers in any way) or simply the top selling products.