For that kind of mileage, it could take you a lot longer to get through it on a more upright bike instead of a road bike. Plus, you are right about having various hand positions at your disposal with drop bars.

If you don't want to buy a bike now for improving your fitness and losing weight and then have to buy another bike later to make long distance riding more enjoyable, then perhaps you can find a touring bike that you can set up into a relaxed position and then you can make it slightly more aggressive (bent over) down the road. Touring bikes are already have the most relaxed geometry of road bike styles, and they are built to hold a good amount of weight. You may find that the right saddle at the right angle will make all the difference, too.