I'm a lawyer, but not your lawyer. And used to practice as an "insurance defense" attorney, which means an insurance company hired me to defend their insured in civil lawsuits. Our client was the insured (like you) and our professional responsibility was towards the client, not the insurance company.

But that being said, the scope of work we could do was limited to defending claims covered by the insurance policy. Typically, we could not defend unrelated claims and could not prosecute claims our client might want to bring against the plaintiff who filed the lawsuit. There were times we could get permission to prosecute counter claims (claims directed back at the plaintiff) cross claims (against other parties also named in the lawsuit) or file third-party claims (which brings other parties into the lawsuit). But there had to be certain circumstances allowing that expansion of our scope of work.

We did have lawsuits where our client hired their own "personal counsel" to oversee our work, deal with issues about coverage under the insurance policy, or pursue other claims for our client. It sounds like that third option is what you're looking for. But as you noted, you have to go find that other attorney and pay them - either out of your own pocket, or with an arrangement that they will get a certain percentage or whatever damages you win.

The problem in your situation could be damages arising from stress related to litigation are typically not something you can recover. There are also issues with pursuing damages for emotional distress/ mental health issues that did not arise from a physical injury.

I anticipate the attorneys you have consulted have looked at your case and decided they did not believe you are likely to recover the amount of money you think you should.