I really feel you on this, being pregnant and having to make such a big decision in such a short time is so stressful, and your worry about the surgery AND the travelling is totally valid, don't feel bad for feeling overwhelmed!
My friend went for keyhole removal of her cyst during her second trimester and she said the recovery was so much more manageable than she expected, less pain compared to open surgery, and she was up and moving around faster than she thought. Someone in my mum WhatsApp group also mentioned that the window between 14 to 28 weeks is generally considered the safer period for this kind of laparoscopic surgery, so if your cyst is still shrinking, it's really worth having a heart-to-heart with your doctor about whether to wait a little more or go ahead.
On the doctor choice, a few mummies I know who delivered at KKH said the access to the doctor can sometimes feel a bit harder since it's a restructured hospital with a bigger team, whereas those who went private said they appreciated being able to reach their doctor more directly, so your concern about post-surgery contact at KKH is something others have felt too, and worth raising directly with Dr Smita before deciding.
As always though, please do discuss all this with your own specialist since every cyst and every pregnancy is different, and they'll know your full history best!