Can a non-related caregiver receive compensation if appointed fiduciary?

The ex husband cannot designate you as a fiduciary or POA, only a Judge can do this. The Judge also sets the compensation rate. What you describe seems more like a Guardian status, you really need to talk to a lawyer.

If your patient is electing you to be his medical Power of Attorney and financial Power of Attorney, don't take that responsibility without some type of compensation made in writing. Being POA can turn into a full time job depending on what all is involved. Don't go into this blindly.

Are there no other relatives that your patient can call upon to be POA? I assume there are no children, or nieces and nephews? If there are, be careful as you might find yourself under the microscope.