It's hard to answer what an insurance company may due. Why not increase your deductable and reduce premiums. Seems like you are prepared to absorb $6000 to keep premiums from going up.

The rule of thumb in any HOA situation is when in doubt - first look at your governing documents. Is there anything that relates to how an insurance claim must be handled? If not, without knowledge of your HOA's financial situation, are the homeowners ready to absorb the $7,000 damages? Where will the funds come from? How will the repairs be handled? How many claims has the HOA filed? The loss seems fairly "low", therefore the chances of facing a policy cancellation are low. Has anyone tried speaking with the insurance agent? There are many variables to explore before deciding not to file a claim and the board needs to consider each one.

Double check in your HOA constitution or your policy and procedures manual. Chances are, something like this has happened in the past and it has been documented.

Your concerns are very valid. You may want to take to your agent and ask for advice. Assuming your agent has experience with similar policies they may now more. Many insurance companies will not raise premiums if this is your first claim in a long time too. 
 
I agree with Dom. It's always better to have a higher deductible to avoid situation like this. 
 
The biggest issue is that any owner living in your building may file a claim against your policy and they don't generally need board's approval. Having higher deductibles helps you avoid it.

A large number of associate members are displeased with some of the current board's decisions and are looking to recalling 3 members. I could not find any reference to procedure in Chapter 183A. How would we go about doing this?

Read your Association's governing documents to find how to recall board members.

We submit ALL claims whether they merit it or not. That is why you have insurance. Not to do so can put your assoc at risk. This is one area where you do NOT guess.