I started on a Cannondale CaadX and just put road tires on it. I was at 336lbs at the time. Never had an issue after I put 36 spoke wheels under it. Now its my dedicated indoor training bike and gotten some more pure road bikes for daily riding after some lbs melted away.
You will be ok with most bikes actually, the wheels are where you need to have the concern. You didn't say how far over 300 you are, and its none of our business but if your well over 300 a road type bike will get a bit more difficult to ride for now.