Recommend Kevin and Willi Motos
I can also recommend and back up Kevin and Willi Motos for bike storage.
Montevideo is a perfect chilled out city to finish or start your trip as well. Close to the beach, beautiful colonial city, friendly people, nice weather, good steaks, 1hrs ferry ride from Buenos Aires, easy to navigate and has international airport connections.
Kevin and the guys are professional and very friendly. They even shouted me a pizza dinner from the gourmet Italian Pizza joint across the road!
The storage facility is 10mins down the road from the Willi Motos bike shop. Its secure and all the bikes are covered. They will even start and run your bike as frequently as needed and will charge the battery also if required.
They even have a small bike museum at Willi Motos which has some classic old and rare motorbikes (photo below), even if you aren't storing your bike you should check it out.