From EWR, check out flyi.com (Independence Air). $49 each-way (more convenient times are higher-priced) but it arrives into IAD which almost cancels out the cost-savings if you take a cab into the city. Nonetheless, this is a good low-priced option for the EWR-IAD route. $179 over that weekend sounds like a steal to me. I just paid $183 (!) for a DCA-LGA shuttle over a non-event July weekend.
I haven't stayed at either property so I can't comment (I've visited both many, many times). Just to keep your options open, don't forget to check out the new Mandarin Oriental in DC. The location is slightly inconvenient for shopping and restaurants but far more convenient to the museums and monuments on the National Mall and hence, the Fourth of July fireworks