If I am not wrong, access to SKL or KF Gold Lounge is only if you are departing on an SIA or Star Alliance operated flight (or one of those SQ partners like Virgin) with travel on appropriate cabin class and/or with appropriate star gold or PPS status.
Scoot does not fall into either category so even if you are booked in J on the long haul but connecting in Scoot in economy or even ScootBiz, you do not get access to either lounge in Changi even if u have PPS or Star Gold status.
can see summary of access here:
https://milelion.com/2022/07/02/sing...-guest-policy/