I know you are asking for something different, but since there are already a few good answers, allow me to instead to reject the premise and give you a different.
It’s not impossible to implement an AI solution within the context your provided. The problem is that it’s going to be expensive. However, you can offer to deliver something smaller, focus on the smallest but valuable contribution you can make. While cleaning up the data is still going to be a hell of task, if the scope is small enough it can be achievable. Then, you can communicate the difficulty to scale due to data issues which can help management undestand the importance of prioritizing data quality.
If you have a bunch of sales data, maybe you can focus on deriving purchase patterns and build a simple recommendations engine. If you want to focus on marketing, you could try lead classification. Ideas depend on the domain of the company you work for.
Become a socialist and realize it’s not your fault, you are just trying to get through.