I was searching for information on this archetype and I had a question that I asked in a thread in the General forum that I realized may better be placed in the Rules forum. So here it is:
Could the ecclestheurge choose an inquisition (which have no domain spells) as their secondary domain and then use Domain Mastery to add spells from another domain?
Yes. A cleric can always choose an inquisition in place of a domain, provided that their deity (if any) grants access to that inquisition. It's generally a suboptimal choice for clerics, but the ecclesitheurge wouldn't really suffer in this case, I suppose.