JOEL - I am not aware of any written policy of the Church that specifies that a member of a Bishopric or Branch Presidency can not wear a beard, unless something very recent has come from Salt Lake. However if you were to ask the opinion of any given General Authority on this subject most of them would probably discourage it.

While explaining the BYU grooming code in a talk that Dallin Oaks gave, he said,
"There is nothing inherently wrong about long hair or beards, any more than there is anything inherently wrong with possessing an empty liquor bottle. But a person with a beard or an empty liquor bottle is susceptible of being misunderstood. Either of these articles may reduce a person’s effectiveness and promote misunderstanding because of what people may reasonably conclude when they view them in proximity to what these articles stand for in our society today."

Of course he gave this talk back in 1971 when in the minds of most people at that time, the beard and long hair were associated with protest, revolution, and rebellion against authority. But that is not really the case nowadays, unless of course the beard makes you look like Bin Laden.
More recently, David S. King, president of the Washington Temple, answering a question in the Feb. 1993 Ensign, said;
"Church leaders, recognizing that fashions go in cycles, are sensitive to the rich cultural diversity within the Church. For example, they have recently held that clean, neatly trimmed and managed beards and long hair for men—as well as certain other fashions that to some might seem “trendy”—are acceptable for the temple, provided they are not inherently offensive or vulgar."

We had a counselor in our Bishopric who always wore a beard and mustache. And as long as he kept it looking clean and well trimmed and groomed no one complained about it. But then we live in California, not far from the Berkely university campus.
I can't tell you what the right answer is. Your husband's situation may be strongly affected by the personal opinion or inspiration of the Branch President or other local leaders there in your area.
Your husband may feel that his beard does not conflict with his calling, but the other leaders are considering what affect it might have on the entire Branch or those who may visit it. He should perhaps ask the leaders to explain to him exactly how they know of this being "Church policy". If they have truly pondered and prayed about this and have been inspired to believe that it is important for your branch where you are living now, then he should try to comply with their council or try to come to some sort of compromise.