Having spent a day and night in Bratislava last week, I can offer a limited observation. I felt that one day was more than enough to see the major sites. The castle is closed, but the exterior is accessible. The downtown area is small and compact worth a couple of hours wander. It is worth it to visit the multi-lingual Tourist office downtown. They are very helpful.

Streets are cobblestone for the most part so schleping the kids may be a factor for you. I stayed at the Sheraton for cash and points which was an ok deal (regular rates are not) It is new hotel attached to a mall.

Taxis are problematic in this city. They want at least $US15 as a minimum from the railroad station or airport. Railway station to Sheraton is 5km. Airport is approximately 20 minutes. Taxi drivers seem to prefer to stand around rather than have a fare for a reasonable price. Meters are available in limited zones.

Downtown is so small that walking is the preferred method.

Despite not driving, the traffic conditions were what one would find in any European country (Italy and Romania excepted). I did take a van from the Bratislava airport to Vienna airport when our flight was cancelled. Note: there are limited flights into Bratislava. In fact only two Czech air flights daily, nine hours apart from Prague. The major carrier is Ryan Air.
One option, if rental rules will allow, is to fly into Vienna and rent there (although Austrian rates are normally very expensive).

Since you plan to drive, I suggest that you check out Auto Europe for cars and to determine if one can take a rental from Slovakia to Split. It is a beautiful drive along the Adriatic coast from Croatia to Split but it will take a couple of days or more. They can also inform you if there are drop off fees within Slovakia should you decide not to drive all the way to Split.