For the distance you are riding and the desire to go faster I would look towards a road bike. Because of the streets you plan on riding on I would look for something that accepts wider tires and have a clyde friendly spoke count.

I can't say for certain why the LBS were steering your towards hybrids or mountain bikes. I think it was probably because of your size more than your weight. 240#'s is not that heavy and just about any bike, road or otherwise would support it just fine. The problem you usually run into is what is available and ready to ride. Since I believe the average height is 5'9 or so most shops will have bikes already built up in a size range suitable for that. Being 6'4 would mean someone would have to build up or order a bike for someone they are not sure will purchase it.

What is required on your part is just a little bit of research. Test ride as many bikes as you can within the budget you are looking at. Component wise they are usually around the same and most work fine without issue. Get a bike you want to ride and you are comfortable on. What you want is something that fits.