Bucharest

I encourage you to continue your look into a lower Danube river cruise. Several river cruise companies offer this cruise - from both starting points. And, the end points (Budapest and Bucharest) are fascinating cities that certainly deserve a longer look than will be provided as part of your tour.
I recommend two (at least) extra days at each endpoint.
As I'm certain you know, Budapest is "on the Danube" - in fact - it separates the old cities of Buda and Pest. Bucharest is located approximately 45 minutes north of the Danube by bus. There is no actual "cruise port" for Bucharest - rather a wide spot on the river with docks and loading ramps near Giurgiu and across the river from Ruse, Bulgaria.

Your cruise company will provide hotel accommodations pre and post cruise - probably in a big-box hotel (Sheraton, Sofitel, Intercontinental, etc.). For your convenience, it makes sense to book additional nights at those locations - avoiding the need to relocate from one hotel to another. However, both endpoint cities have numerous boutique hotels or small family run locations with wonderful opportunities to taste the local culture more personally.
My only experiences in this region have been in spring - and the weather was warm and sunny - but snow had fallen not a week prior to my arrival in Bucharest (April, 2018).
I would not pretend to know "which is the superior river cruise company" other than to say - some are three times as expensive and others, some offer remarkable fares including air - and some focus more on culture and personal contact.
I can expand on that if you wish, but the search for river cruise selection typically begins with available dates and cost.

The cruise dates we are looking at are 3rd week in May or last week in September

I'd suggest taking a look at the actual, historical, day-by-day weather statistics available for many cities on the website timeanddate.com. Look at your prospective weeks for at least the most recent three years (I'd go with five, myself). See what you think. Sight unseen, I'm going to guess your May dates will be warmer (perhaps too much so, depending on your temperature preferences) and drier than your September dates, but I could be wrong. Within recent years I've only been to Budapest once (May 2018) and Romania once (early September 2015). That's why years' worth of actual historical data is so useful. Those of us who visit an area for not much more than a week can have been there at a time when the weather was doing very odd things that are likely not to be repeated. I've linked to Bucharest's weather data for September 2019 on timeanddate.com.

Budapest is a great place. Bucharest, frankly, not so much. I wouldn't tell you not to spend additional time there, because I'm sure it must have some interesting museums. But architecturally, much of it is a mess because Ceausescu tore down so many of the beautiful old buildings. Prettier cities, should you have the time, include Sibiu, Brasov, Cluj-Napoca and Timisoara. The small town of Sighisoara is very picturesque. Unfortunately, transportation within Romania is rather slow (there are few trains, and they are not speedy), so you may find it impractical, with only two days, to see anything other than Bucharest.

While Romania is a fascinating country, with only a few days available, I'd recommend using the time to Budapest (or day-trips from that city). Go back to Romania when you have time to spend at least a couple of weeks in the country.

I have read that the Bulgarian town of Ruse (mentioned by Blue439) is attractive. Unfortunately, I came across that information after I had passed through it on a train heading south to Veliko Tarnovo.

I enjoyed Bucharest. Good for 2 full days anyway. Budapest is a 3 full day minimum and 4 makes more sense. The boat docks in the center of Budapest so that's no problem. I dont really see the beauty of the Danube downstream from Budapest, but each to their own.

Date? Definitely late September. Beautiful time of year in the region.