If you want them to work on a holiday, you are probably getting premium rents on those days and you should pay them extra.

How should I expect the premium to change?

She's willing to pay a premium for the month-to-month arrangement.I've never done a month-to-month lease before and I'm worried it might be a hassle, especially since I don't know how the market will be in the winter.

Markets are getting more sophisticated and truly pricing in a lot more premium on any angle you take.

They got me a quote from another provider and it was about 5x my current premium.  

-- can this be altered if a business pays the premium? 

One thing that may help is replacing the roof - a lot of companies either wont write or overprice premiums for homes with roofs over about 5-8 years old because it's a replacement waiting to happen with the next hurricane.

The Assessments levied by the Association shall be used exclusively for payment of Common Expenses including, without limitation (i) the upkeep, maintenance and improvement of the Common Areas and other maintenance obligations set forth in Sections 10.1 and 10.3 below; (ii) maintenance, repair, replacement, and operation of rights-of-way and easements within or immediately adjacent to the Project (e.g. landscaping and sidewalks within the right-of-way of adjoining streets) to the extent that such actions are required by government entities or deemed appropriate by the Board, (iii) promoting the recreation, health, safety and welfare of the Owners and other lawful occupants of Lots within the Property, (iv) payment of property taxes for the Common Area; (v) insurance premiums; (vi) legal and accounting fees; (vii) management fees; (viii) charges for utilities and other services provided to the Common Area by the Association, if any, and for all utility charges paid or incurred by the City and reimbursed by the Association for street lights in the Common Area; (ix) costs to maintain the entrance features, including any electricity, landscaping and irrigation expenses associated therewith; (x) the performance and exercise by the Association of its rights, duties and obligations under the Project Documents, and (xi) maintaining any Stormwater Control Facilities located on portions of Common Areas to the standard required by the governmental entity or agency having jurisdiction over such areas.

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Instead of filing a claim against your insurance and having it cancelled or your premiums raised, it goes against the renters insurance.