It sounds like you really don't want to/have the time, but I really recommend going the VPS route or dedicated server route (generally more expensive). As with my experience, any normal shared hosting will just take you for your money and run with it, and still limit you.
A VPS service or above would be better. I used to use stormondemand.com until I shut down my main website that was using it. I have no idea how much traffic you seem to get, but mine was just using a 4GB VPS server slice easily with all the simultaneous traffic it would get at once. But I'm sure your's would be much bigger resource then mine. At any moment at my peek, my site used to get 10k to 50k simultaneous hits. But, my site was more of a relay sort of site (think of a 1-hit spam server... :lol ), so I heavily optimized my server to reflect this, even had my DB as static stored in Memcache. I would randomly guess your site would be more heavy on traffic then mine ever was. I recommend looking into Storm On Demand.
You mentioned you already have a server, so that kinda takes you a step forward. Did you buy the server and it's hosted somewhere else? Or did you buy it and it's in you're own physical location?