I know this question is old, but here's one thing I think I must add.
It sounds Mormon. I can bet that it's Mormon. And my translation is:
He's a president, but he doesn't have priesthood.
In a modern Korean context, the most probable situation where one can hear "신권" and "회장" in the same sentence would be Mormon (or of the Latter Day Saints).
Maybe this is too much of a personal experience (skip this paragraph if you don't want to hear any), but I never heard the two words in one sentence outside of a Mormon context. (Not a LDS myself, but have some friends in the Korean LDS church.) Actually I think I never heard the word 신권 at all before I met these Mormon friends. I was born a Presbyterian, now Catholic, but that sentence sounds very unnatural in either side of Christendom. Only Mormons would imagine such a sentence.
In a Mormon context, 신권 is the translation for priesthood (I won't be discussing the concept itself here, but if you need an English reference, see link), and 회장 is the translation for president (English reference). Both are everyday words of LDS missionaries. I think the LDSs would understand "회장" in a neutral context as the president, the guy at the top level of church hierarchy (like a Catholic pope), but it may refer to any president of various sub-organizations in the church. (Correct me if there's a LDS in the community.)
I think LDS theology (or church structure) requires one to have "priesthood" in order to hold most church offices called "presidents," so maybe the speaker of the sentence is casting doubt of his authority.
Another point supporting my theory is the use of the postposition "-로되". It is one of the words that are being more and more outdated (or should I say archaic). But it is widely used in the New Korean Revised Version (개역개정) of the Bible. I met some foreign missionaries in Korea who learnt Korean from that translation of the Bible, using the word "-로되" in colloquial speech.