Thanks everyone for the quick responses. At least I have some comfort in knowing I have an option if I can't sort out a visa in time for my trip on the 16th.
One further question - given it appears a nested trip to Taipei, Jeju or Macau is a fully legitimate option, would it be safe to do a same-day return? That's the implication from your responses (no need to stay overnight and back-to-back TWOV is allowed and won't raise an eyebrow) but just wanted to triple check.