I think it's unreasonable to expect OP to pay more (either miles or cash) to fix an issue that was fully SQ's responsibility.
If this was a revenue fare, we would expect SQ to accommodate the OP on another reasonable itinerary.
My view is that the OP should ask SQ to:
1. Open space on one of its SIN-USA flights in the ticketed class of service. This should not entail additional cost or miles.
2. If there is a code share flight into IAH, book positive space in the ticketed class of service. Another option could be a UA flight if award space is available - I don't expect SQ to pay for UA positive space (though I think they should). If not, OP may wish to self-arrange this domestic leg to avoid having to quibble with SQ.
Given SQ's dismal call center with minimal employee empowerment, this may require submitting the online form. I don't agree with the suggestions that OP should be on the hook for extra costs, this is clearly an SQ caused issue and the airline arrange for alternative flights to fulfill the ticket.