Your budget is definitely low, unless you find some black box server with who knows what brand of hardware inside on a shotty network.
I'd recommend increasing your budget to increase reliability in network and support. Go for an enterprise provider and don't make the mistake the first time.
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If your willing to wait a little bit DreamHost is bringing back dedicated servers very soon which should be in that price range. Likely going to be half the amount of ram though. It will be 100m unlimited bandwidth and in Los Angeles. Should be powercycleable from the panel and its managed.
I don't find the budget to be all that bad, certainly something you find among a lot of decent (or even great) providers. A dedicated 10mbps uplink is something a lot of providers will be willing to give you. Definitely contact webnx - they can probably work out something in the vicinity of your budget. You should contact their sales department directly (not pm on this forum).