Thanks for writing about this question. I think it is an important one.
The idea of seeing one therapist has been around for a long time, mostly so there would not be conflicting feedback, or more deeper concerns with transference, or the possibility that the relationship usually needed to make the changes in your life would be thwarted by a second therapist.
But the thinking and the practice of this has changed radically in the last 2 decades. As an example twelve-step programs and therapy were estranged, now they are more likely to be seen as mutually supportive interventions. Couples therapy needs to be done with a different therapist than the individual, group work is separate from couple and individual, and trauma interventions may require specific treatment.
Therapy has also morphed into a very different form than it was before managed care and other insurance matters. Therapy can have a highly specific focus, with specifically trained therapists who work with you on specific problems. I regularly invite my clients to see another specialist with a particularly different skill (such as EMDR) when they need it, and other therapists will send their clients to me when they need my specialty (For instance, group therapy).
As far as the “service extender” status this is something no clinician I know understands. The core feature seems to be the insurance companies use this phrase to suggest a duplication of services. When this has happened I know the insured and the clinicians have written to substantiate the fact that it is not a duplication, but a different service. When I have to do this for the purpose of explaining the difference between individual and group therapy the insurance companies sometimes change their position. My suggestion is to challenge the insurance company decision if you feel strongly that the two therapies are different. The worst that can happen is the insurance company will deny it.
So the short answer to your question is yes, you can. The one thing I would suggest is not to keep it a secret from either. Be clear and upfront about what you are doing and why. You don’t want the process of getting help run the risk of you feeling bad.