Getting around Romania is somewhat challenging by public transportation. The trains, where they exist, are not faster than buses; I figured I covered about 30 mph back in 2015. Back then I got myself to Suceava after somehow learning of a tour of the painted monasteries departing from Suceava. That particular tour, organized by a local travel agency, wasn't outstanding; I had a feeling the young guide was a fill-in, but the tour got me to the locations of several of the monasteries.
I haven't been to Iasi. I though Cluj Napoca was very pretty (as are Brasov and Sighisoara). The Retro Hostel in Cluj has a large slate of tours it organizes. You don't have to stay at the hostel to take one of the tours. At the time of my visit, many/most of the tours didn't run on a fixed schedule, but rather on demand. I was thrilled to hook up with one or two other people who wanted to tour Maramures, the NW corner of Romania where the wooden churches and Merry Cemetery are located. It was a really wonderful 2-day tour with great variety of sights and quite inexpensive at that time. We spent the night at a rural B&B where we were fed 3 meals primarily consisting of products grown/produced by the owners, including their own cheese. That tour remains one of the top highlights of all my (long) trips to Europe. I recommend it highly if it's available for your dates and you can't make Bucovina work.
I see Retro Hostel has a 2-day tour of Bucovina as well as still offering a 2-day tour of Maramures. There's also a 5-day tour of both regions, but it sounds as if you probably don't have time for that. There are many other options.
I'd recommend contacting the hostel with your dates to see what tours are set to go already and letting them know you especially want to see Bucovina. They say the tour just requires two people to go, so you just need one other person to express interest. (I assume they'd be happy to arrange a solo trip; it would cost you more, but it might still be extremely reasonable, because Romania is a low-cost destination.)
My tour used the driver's personal car (comfortable and air-conditioned). I imagine tours with more demand use vans.
https://www.retro.ro/retro-tours