Once you reach SIN you contact SQ transfer desk or lounge to let them know that your bag is on the belts from the Scoot flight. Sometimes - especially when you are in J/C - SQ will agree to take your bag(s) from the belt and tag to next destination and you can remain airside. It is worth a try although entering SG to get your bags from the Scoot flight and check-in at SQ counters, should be no issue. No doubt that your request to tag bags to JER on separate BA ticket will have a higher chance of being succesful if you show up personally at the SQ check-in desks at T3.