I know that what I offer as an opinion won't make a bit of difference, but here are my thoughts:
1. Pre-baby my wife and I road-tripped a lot, and we still do occasionally. We've done ~800 miles in a day, but it's a long day(12 hours or so for the trip we've taken) and at that point you just want out of the car. That's also with us taking turns driving. When I punch Washington DC(I know, the OP is in the suburbs, but close enough) to Waterloo, IA in I see that it's around 16 hours. That's a LONG day, and it's really a long day to have to turn around and do it again the next. 32 hours of driving in 48 hours doesn't sound appealing or safe to me.
If I were to do it, I'd do 10-12 hours the first day, overnight, then drive the rest of the way in to pick up the part(arrive when the yard is likely open...) then get back on the road, go as far as I reasonably felt I could, and overnight again before making it back. Yes, that's a 3 day trip and 2 hotel nights, but it's the only thing I'd really even consider for my safety and sanity.
2. The cost calculations presented here don't take tolls into account, which Google Maps tells me there are through Ohio and Illinois. I know there are some toll roads around Chicago. Without bothering to look up the rates, that's probably going to add $20-30 each way to the trip. I tried to look at my Easypass account to see how much it was the last time we passed through Chicago, but couldn't find it.
Yes you can avoid tolls, but by my counting that would cost 100 miles in each direction and 2 hours. I'd care more about the time than the miles on a trip of that length.
3. All of these time figures are assuming things are perfect. Time passing through Pittsburgh, Cleveland, and Chicago are all going to be very dependent on time of day, and hitting during rush hour could easily add an hour or more to what it takes to get through at 10:00 at night. Even smaller cities like Dubuque can clog up, too. I'm guessing you'd hit rush hour for at least two of those each way, and now you've turned your 16 hour each way trip into 18 hours.
Rush hours are sort of predictable, but other traffic delays aren't. There's construction everywhere this time of year, and don't forget that traffic slow downs can be a self-perpetuating cycle as they can cause accidents that cause even more slow downs and the like. Not too terribly long ago, we did St. Louis to Nashville in an evening(first leg of a two day trip). That's normally a ~5 hour trip, and we thought we'd miss the worse of the traffic leaving after I was done with work for the day as we did. All said and done, it was 8 hours because of construction delays, accidents, and the like.
4. Maybe I'm too picky about hotels since I like things like clean sheets and covers without too much DNA that's not mine, but these days I'm thrilled if I find a hotel I'd want to stay at for less than $100/night. Don't forget too that most hotels post their rate before taxes and fees, which typically add 10-30% depending on locality(many places have taxes on hotels over and above sales tax). As an example, a recent one was $169 advertised, actual billed price was a few pennies over $190(around 13% for anyone keeping track). The last sub-$100 hotel room I got was in middle of nowhere Missouri at an older, non-chain motel that was clean and well kept but nothing fancy. It was more than fine for the stay, but you'd not have found the hotel unless you'd planned to travel to the area. To be honest, a $50 a night room...I'd rather sleep in the car
5. I've done a fair few road trip miles in my life, and on the whole it's pretty rare to have something happen on the road especially if you're in a modern-ish reliable car, but it can still get pricey if something does happen. Don't forget that you don't have your own tools, garage, space, etc and even if you're handy, you're probably still limited in what you can do yourself. You're also potentially in an unfamiliar area where you don't have any idea where to go to have something fixed, may pay through the nose for a rush repair, and may still end up having to add an extra day or two if you need to wait for a part or just to get into someone who can fix it. Last summer, I was on my way home from my parents' house in Kentucky and lost my alternator about 30 miles from home on the way back. I took my chances and made it home no problem, but if it had been further I'd have needed to get creative or just suck it up and either have it fixed or towed. There wasn't any chance I was changing the alternator in the MKZ in a parts store parking lot(it was bad enough at home...). I have a fair few road trip miles on this vehicle and this is the closest I've personally come to having anything go bad in it, but it does happen. It wasn't too terribly long ago, though, that my mom ripped a hole in a tire(actually in this same MKZ-back before I bought it from my dad) on Jellico Mountain in TN...although that one was just a few hours detour at Wal-Mart.
Long and short of it is I'd take the trip if I had the time and wanted to do it. Heck, back in 2020 when I was on furlough from work I nearly did a trip to Philly to get an MG OD transmission someone else wanted because I had the time to do it and wanted to visit with the person it was coming from-I offered to do it for the person for my cost of gas only(which was about 2/3 what Fed-Ex would have charged) because I wanted to do it, but under other circumstances I'd not have even offered. I'd also do a trip like this if shipping weren't an option or if I didn't trust safely shipping. I wouldn't do it purely to save money, however, even if on paper it looks like it would.