If I was in your position - I would push for integration of some sort while keeping the two products separate for now. Then have a separate team from both organ

If I was in your position - I would push for integration of some sort while keeping the two products separate for now. Then have a separate team from both organizations start on a new product that borrows from both products. Obviously it’s easier said than done - ego have to be put to the side. The advantage is you keep the status quo with minimal customer and cash flow distraction while working on integrated solution.