Would it still be binding if you chose to break the lease and move out
the following month when the same problem persists?
At first glance it seems a bit stretched, but only you know the full contents of the landlord's message.
The key issue is whether "end of the month" in that message can reasonably be interpreted as "end of the month in which you intend to move out". If so, the doctrine of contra proferentem would apply and favor your position, since you are not the draftsman of the condition that you and the landlord agreed upon (provided that you expressed your agreement).
Obviously the most straight-forward, most certain option is to inquire of the landlord about the validity of this condition in the following month. There are at least two arguments to support your intent: (1) Moving out by the end of the following month results in the breaking of the lease being less premature; and (2) the landlord would have longer time (namely, one additional month) to find a new tenant.