1. having a written lease agreement would qualify you to purchase rental insurance with additional floaters available for firearms valued over the initial insured limits ... for me, the initial homeowner's insurance (State Farm) will only cover $3K of firearms, that is why additional floaters are so important ... this type of insurance is sold on an annual basis so the insurance cost if the lease is canceled at the beginning of the 'month to month' rental would seem unusually high
2. from your description, the 'month to month' lease would be brief and informally written and your permittable activites, such as firearm storage, would probably be a moot issue (an issue not addressed in the lease)...having the 'landlord's verbal permission' does not guarantee that your activites would be permitted the next month...a well written yearly lease detailing everyone's rights/obligations would be in your best interest
3. you identified many people who could have access to your (presumably unlocked) room...boarders in other rooms, landlord, children, etc...even if you had your safe, insurance and verbal permission from the landlord, i would not store my firearms there
4. the only viable alternative seems to be a 'secured storage building with environmental controls', but this may not be economically feasible, because it would also require a written longer term lease and also supplemental insurance
YMMV
good luck