Out of your 3 options I would go with a variation on number 3.
3.Remove the button, and leave the box clickable. This gives user more clickable area.
I think having 2 buttons going to the same place is confusing. And the act of clicking a tile/box to navigate to the primary action is very common place on interfaces these days. Especially if there is only one action.
Variation is: I would remove the inner button but maintain the text. The text could be a helpful indicator of what tapping the tile will do if its not already clear, if it is clear then remove it altogether. You may need to tweak the styling of it, e.g. remove the box and make the text blue. It would act a little bit like the action text of the whole tile (if you keep it).
If in the future you decide to add other actions that live as inner buttons to the whole tile then that would be fine (see android cards https://www.google.com/design/spec/components/cards.html#cards-usage), as long as they're styled to look obviously separate actions and have enough outer space.
N.b. you say its not for touch based environment but there is no harm in considering how your app may evolve and become so. So factoring that in at this stage will help the evolution into other platforms easier and also current users will transfer their knowledge of how to use your non-touch app into a touch based environment more easily.