I have done something like this, but my pitch began before I got hired. So during the interview process I explained that I am involved with a side business and

I have done something like this, but my pitch began before I got hired. So during the interview process I explained that I am involved with a side business and that the product is a useful tool for helping me with the day job I was getting hired for and asked if they would open to me using it for my team. They said yes and they've been using it since. Your situation probably requires a more delicate approach but really it will depend on the culture of the company. Not every company is a massive bureaucracy. There's probably more chance if it's a smaller company that just wants the best tool for the job and deliver results, and if that's your solution that may be fine.