We go on day trips to DC and I find that the larger the group, the more than metro and meals become a budget buster.
I try to avoid the fast food, but have been desperate enough on a weekend to suck it up and it at McD's as the town operates for the Federal Workweek and not a
disneyland tourist attraction. Kind of annoying, but totally understandable.
For a non-fast food, Smithsonian meal with NO dessert and each person having their own plate, our family of 5 pays about $60-$75 depending on what is selected. I love the art museum cafeteria and the cafe at the Museum of the American Indian (I struggle to get that name correct!). The latter tends to be pricey, but worth a trip to American Indian inspired food. We attended a cooking demonstration once and they said that they rotate their menu every season trying to utilize what is in season. Meals are not authentic to any specific period, but are influenced by ingredients that may have been used.
The price I quoted is for a non-sit down establishment. If you have a served meal, expect to pay more.
We ate at a pizza place near the International Spy Museum and spent about $100 for our family five. Certainly not cheap, but when offerings are restricted, we just do what we can.
We often will bring a meal with us and as soon as we get off the metro and out of the station, we will sit somewhere and eat our "picnic" lunch. We also carry snacks. This way, it cuts down on eating out costs--so I would totally suggest that.
You cannot eat your own food in museums, but we have realized that you have sealed food, you can at least carry it with you. (I had misunderstood policy as absolutely nothing could be brought in--but you just can't bring a cooler nor can you eat your own food in the museum.)
To do DC on the budget, avoid the paid museums and stick to the Smithsonian circuit and traditional site seeing.
One thing I discovered when passing along tips--there are FREE concerts EVERY DAY at the Millenium stage at Kennedy Center. You can view current schedules for upcoming trips on their website. They have the month of May. I am not sure if June is up. I do know that the rest of summer is not up.
For metro--to save some cost, be aware of when the busy times are--rush hour = higher cost to ride! Don't take metro every chance you get either--be willing to do some walking! If you are staying in the city, you don't need the metro all of the time. My 4yo son walks great distances! We do bring his stroller as he is heavy to carry when he is tired. But he lasts quite a long time. My 10 and 8yo don't have issues unless it has been a really really long day. We just make sure that they are dressed for the occasion of walking. It is not the day to wear the prettiest dress shoes.
Also--it is raining, but you won't melt! Bring a poncho, wear an umbrella--it will be okay! Even if it rains all day.
Since you have a wheelchair, be aware that elevators for metro are often situated a bit near the regular entrance of the station. Sometimes it is not obvious and around the block or something. Sometimes, for a stop with multiple entrances, it is only on one side. L'enfant plaza is confusing!!! It is multi level. I followed the signs after we came in on the orange line. Took the elevator up and then went to the closest exit. Ran my card through and was out....it was then that I realized, there was no elevator and they don't really let you back in. Scan the exit for an elevator before you run your card. If you don't see it, double check with the station attendant. We had to go rogue and do the escalator with the stroller. Totally against the rules, but I was by myself and we were the only ones around.

I hated doing it. (prepared for the smackdown from other locals.)