Don’t do it! Applying without LORs with an incomplete application would be a disaster, even with good scores. SOAP is WAY more competitive than Match, in Match you have 2:1 applicants to positions ratio, in SOAP it’s 20:1, and you’ll be competing with people from ortho and ENT who didn’t match to their highly competitive specialties and now go for anesthesia with excellent scores, experience and recommendations.
You have terrific scores. If you shadow some anesthesiologists to get US LORs, or work at an academic department to get some publications/recommendations, given that you don’t mess up your interviews you’ll land a position at a top-10 program. Why scramble for a random spot now, when you can work on perfecting your application, taking step 3, and land a great position a year later?
If you’re dying to start practicing, apply to SOAP for a prelim year with letters from your country (transitional year, IM or surgery, I highly recommend opting for a transitional year or cushy IM, surgery prelims generally suck). Get the letters of recommendation there and apply to NRMP in the proper cycle.