@milgreen
What are your planned dates for May 2027? If you do rent a car there are certainly many spots to stop and visit. For ex. Plzen is home to the other mother brewery Pilsner Urquell. The town was the only region liberated by the US. By chance the dates coincide The 81st Liberation Festival wll be May 4-9. https://www.slavnostisvobody.cz/en/ Very Pro-American town!!
There are many castles to visit, World Heritage sites. natural areas etc. Between CB and CK is Hluboka castle.
Some random thoughts.
The atmosphere in Karlovy Vary is great but I find the evenings a bit boring.... restaurants are lacking. It is great during nice days, meandering the length of the collonade, visiting the thermal pool, nearby Lockett castle, finding homemade spa wafers. Honestly I've never felt a big need to stay overnight but if you want a little down time why not. (There are two other spa towns in proximity to KV which create a triangle if you are interested in a spa stay)
Cesky Krumlov is my favorite go to place. 1 night is sufficient but again 2 nights if you wanna take it easy. Definitely get tickets for the Baroque theater. (If you have time you could do the castle tour but it is not of the essence IMO). Pappa's Living restaurant, find the Gypsy Bar for music, visit the gardens beyond the theater,
In either case, you could leave your bulk luggage in Prague then just take an overnight bag if you decide on public transport. RegioJet bus is the first choice for bus service.
I've only ever used public transport while in CZ as public transport is quite good and affordable.
(I am from the US living in Czech Rep. You are welcome to DM if you need any assistance.)