9 times out of 10, the bouquet stems that you see at the bottom of your bouquet are not real. The flowers in the bouquet have a hidden water source in the center, and they add the faux stems to preserve the 'look' of the bouquet. This mean that keeping your bouquet in water may not help.
I think having the refresh in your budget is a good idea. They keep that flowers at the right temperature, and replace any that have wilted too much. I think it's money well-spent no matter what, since it is one less thing for you to worry about. In the grand scheme of a wedding budget, it is really a small expense.
Before you decide to cut it from your budget, carefully consider the types of flowers you will have in your bouquet. For example, if you have a lot of hydrangea, that flower wilts very quickly and therefore the refresh is actually cost-effective. If you have the type of flower that will last a long time naturally, then maybe the refresh is not money well spent. Keep in mind, however, that once any flower stem is cut, the life of the flower is decreased as well.