I stayed in Split recently and am returning to do some climbing.in the Autumn. I can't comment about the climbing, but feel sure you will go to Omis at some time.
Don't think of Kalymnos but picture very lane dependent city driving and a busy coastal strip leading to Omis with lots of beachfront apartments and sometimes difficult parking to access crowded beaches. (30 mins by car). There are lots of accommodation options and with a car you could easily split (excuse the pun) your trip between a few days near Paklenica and a few days near Omis.
It is likeley to be vet hot.
Prebook and hire a car from the airport would be my advice as hiring anything once you are in the tourism oriented areas is much more expensive. Take the full to full fuel option and think about getting an annual excess hire insurance from the UK for around £40 too. Hire companies can feel very threatening, but expect to be charged for a full tank of fuel (refunded eventually) and a huge amount (£1000+) blocked on your credit card as a deposit against damage until you return the car. In Split there will also be a charge if you intend taking the car over to one of the islands or cross a border.