Just to let you know that Gazipaşa is located even souther compared to Alanya so as you mentioned you would at least need a 2.5-3 hour drive to Antalya. A taxi is likely to cost about ₺500-1000 ( max: €65 ) - not sure about the updated taxi prices in Antalya so it's up to you.

If you are going to Manavgat, Side, etc... I would have considered doing what you said but if you are going anywhere west than Konyaaltı, it would be better if you fly to Antalya instead.