IMHO if you are maiking a sme day connection back at SIN you should check either yourself OR yourself and your bag at SIN outbound for CGK-SIN-MNL. Consequentially:

1.You would be issued with boarding cards for all sectors outbound at SIN.
2. Your bag would be tagged CGK-SIN-MNL and would have to travel this route with you. I do not believe that the Agent would be able to check your bag just SIN- MNL unless you were doing an early check-in on a separate discreet ticket for this route. Remember also that the check-in Agents, unless you are able to use the F check facility, are NOT employed by SQ but CIAS, so there would be less potential flexibility. Not that SQ ground staff are renowned for their flexibility either! Another alternative, using only the SIN-CGK and checking the bag to CGK only would not give you enough time at CGK to clear Immigration, claim you bag and re-check. This would also cost you a USD 25 Visa Fee (Unless you have an ASEAN Business Pass or are an Indonesian Resident) AND an IDR 150K AIrport Tax.
3. In CGK arrivals go up the transit escalator on the right before going down the ramp to Immigration and head for the SQ lounge. In the lounge, or at the Reception desk if you are flying Y, ask to speak to the Station manager, explain the situation and ask him to confirm to you either in the Lounge or at the Gate that your through bag has been located and is on board the same aircraft back to SIN and MNL. If you want to be sure, don't board until you are given this confirmation.
4. At SIN on the way back you would be in Transit only and would NOT have to go down to the Immigration Hall and would NOT be subject to Airport Tax. Your "Transit" security check would be at the Gate. The final alternative, leaving the bag at SIN in Left Luggage then collecting it and checking the bag to MNL, although you would theoretically be subject to Aiport Tax at SIN, you would already have your through Boarding Card to MNL and, in my experience, in similar but not identical circumstances, I have never been challenged to pay, I guess mostly because at SIN it is always included in the ticket price so neither Check-in staff nor Immigration are charged with collecting airport tax.

IMHO, checking the bag with you all the way gives you the maximum convenience whilst, recognising that CGK is not the most efficient, ensuring that your bag gets turned around and back on the returning flight and on to MNL.