To be honest, to find a 4 BR, with good DCPS, you would like to be west of the park. Not certain $650 gets a 4BR anywhere west of the park though. Just doing some looking, it looks like Shepherd Park is actually quite good. This means being north of Alaska Avenue. If you can buy up here, your kids would go to Shepherd Elementary, and then to Deal Middle School in Tenleytown. Both are good. The assigned HS is Coolidge, so you would need to find a charter, or hopefully it improves by then, but that is a long time from now.
Other options are Barnard ES. This is the best ES in Petworth, which to get a 4br, is ultimately where you are probably going to have to be. Easy drive to Bethesda, which is where I commute to daily from Petworth.