Can I appeal an unfavourable INIS decision reached in error?

I am an American who has legally been living in Ireland since July 2013 under a Stamp 3 permission to remain. I entered the State under a Religious Volunteer permission to remain, which gives a person the option to request an extension of permission to stay in Ireland after three years (my eligibility to request that extension was confirmed by a GNIB immigration authority).

I submitted my extension request to INIS in February and received a response at the beginning of June that it has been denied because the INIS authorities argue I entered the State as a Lay Volunteer, and Lay Volunteers can never stay past three years. I have a passport stamp that shows I entered the State as a Religious Volunteer and have provided that evidence to INIS, but they are still holding fast to the idea that I am, and have always been a Lay Volunteer, and therefore, I must leave the country at the end of August.

I know that if I stay beyond my current permission, I would receive a Section 3 letter about deportation, and that I'd have the option to appeal that, but does anyone know, is there a way I can appeal the refusal I've been given? If so, how do I go about that? I'd much prefer to never get to the stage where I might have a deportation order on my record. I am completely self sufficient in Ireland and have private health insurance. I've complied fully with my permission to remain, and besides, have volunteered over 4,000 hours since arriving here. Yet, I am very frustrated and discouraged because trying to communicate with INIS to try and clarify things is like talking to a wall.

Thanks in advance!

Sorcha