You don't need to exercise your option to hedge your long eur position. You have already hedged it by buying a put. Your maximum loss is the cost of your put. When the eur drops and you wanted to unwind your postion, all you need to do is to sell your long eur and sell your put. You will make money on your put, but lose on your futures. The total loss will be less then exercising the put option due to the time premium remaining in the put option. If you decide to hold onto your postion, you can't lose any more than spread between your eur purchase price and the put strike price.
A reason for using a put option as a stop loss is that you have the choice of closing out your eur position and hold onto the put if you expect further downside movement.