This is most likely not an issue. MI allows early check in up to 48 hours in advance. There will always be someone at the transit desk in T2 (MI's terminal) because MI is a SQ group company - can't imagine the transit desk for SQ ever being unmanned. Worse case (assumes you can enter Singapore without a visa) you enter Singapore legally, go to check in and use the Self-checkin machines to get the boarding pass. Or check in online before leaving home. I think you're well covered regardless of what LX does/doesn't do.
Changi Airport |SilkAir