I think it depends: if you do condo, will you be in a neighborhood you prefer for schools and thus want to stay in longer (noting that in DC places with higher-regarded schools have a lower chance of you getting a PK space)? Do you like DIY or will you be happy not to have to deal with roof, gutters, windows, landscaping, etc. while you are also dealing with young children? Is there any chance you can find a 3br condo you like, for example
https://www.redfin.com/DC/Washington/3930-Connecticut-Ave-NW-20008/unit-501H/home/145670981 (a coop, so the fee includes utilities and taxes) or
https://www.redfin.com/DC/Washington/4201-Cathedral-Ave-NW-20016/unit-405E/home/9951663 (I do think the fee here is ridiculous and the unit should be cheaper given that)?
There's a house on Primrose St. in Shepherd Park that if you can nab it would also be good and in your price range, but I wouldn't rule out a condo if that works better for you.