The broker then files it with the old carrier & we move on, sometimes we get a response, maybe even a check for what's outstanding & sometimes nothing.

as your finding out wholesaling is very tough .. if you want to get into real estate you may want to look at getting in the industry by being licensed then work at a brokerage that does what you want to end up doing in your Carrier. 

I would shop your price with a broker that has access to 30+ carriers.

Also, you may want to find out who the Associations insurance carrier was and find out from the Adjuster who handled it why her damage was denied. 

There are absolutely options, albeit limited - I can think of three carriers, and yep, all thru broker.

What carriers do you recommend? 

Got a ridiculous quote from my current homeowner carrier (State Farm) for $10K annually (seller is currently paying about $2800 annually). $10K will certainly throw all my numbers off track.

In terms of the property, it depends on how many units and location (among other factors)  to know which carriers will be competitive while insuring your investment.

Do most big-name insurance carriers offer Professional E&O policies?

Some carriers won't list the LLC as the primary and has to be on as additional insured.