You simply have run into an incompetent or unwilling agent. As SQ and A3 have an interline e-ticketing agreement shown as in effect today. it is definitionally possible to issue a single ticket.

However, rather than creating a scene with SQ which is more than likely to end poorly, I would simply have a competent TA issue the tickets and pay their fee (by TA, I do not mean an opaque vendor commonly referred to as an OTA). Perhaps your employer has one on retainer or can recommend one.

In any event, if you are on separate tickets, you will need to allow sufficient time for the transfer between tickets as you will not be protected should you miss your onwards ticket on A3. Thus, there ought to be time to recover and check your bags, hassle that it may be.