I think you're in quite a tight spot here...
The reason I say this is because unlike GMAT, TOEFL scores are not available immediately. TOEFL scores come out 3-4 weeks after you have appeared for the test. The same amount of time is also required to report the scores to any of the schools.
I'm reasonably sure that schools that have a stringent requirement of TOEFL will not accept an application without a TOEFL score. I think you should -
1. Get in touch with your target schools and try to find out if there is any way in which TOEFL can be waived for you
2. If it cannot be waived, explain your predicament to them and request them to consider your application with a delayed score (note that even if you do appear for TOEFL in mid-Jan, you would get your scores only around mid-Feb)
3. If all else fails, apply in Round 3 or if you have the will power to wait it out, apply in R1 next year...