I don't know what the plan is for using refillable mugs during the refurb, but you can ask at guest services when you check in. Fresh serves Starbucks coffee, so the plan may be to get the mugs refilled there.
As for inexpensive breakfast options...there is a breakfast bar set up in the lobby of the Swan each morning where you can get coffee (not the refillable mugs normally, but maybe during the refurb), baked goods, fruit, etc.