I think that your best bet would be the Master at Arms (MA) rating in the Navy. MAs are eligible for two special NECs (Navy Enlistment Classification Codes); NEC 2005 – Dog Handler, and NEC 2006 – Kennel Master, though the latter is only available to E6 and above.
The Air Farce also has dog handlers, but those pretty boys can’t fly their toys. Ever see one of them try a carrier landing? It’s pretty amusing :D They sure know how to keep their planes shiny though.
As for your eyesight barring you from being an MP, is your eyesight correctable to 20/20? If so then that is good enough for MA.
However, I think it safe to say that those dogs are more for security wirk than for SAR operations, or at least that is what they usually are assigned to since there is always need for security but it’s not every day that they need dogs to find people in rubble or wandering in the woods.