The FAA are very uneasy about this, the new york FSDO have always been the the first port of call for such applications since 2006 when the requirement for an "issued" type. Since then the Feds have been policing this because up until recently insurers were not.
Expect to get blown off in NY even after speaking to somebody there and you setting off with your paperwork in order, to even have a remote chance of success you will need to have evidence of groundschool, an oral and paperwork demonstrating that you have done an internal course at an operator of this type (to include night circling approaches in a lear............!!)
Even then they might send you away emptyhanded. None of this is in the 61.55 regs, the days of wandering over to a flightschool and selecting the guy with the shiniest shoes to do three followed thru touch and go's are over.
Within the last 12 months insurers in the EU area are asking for evidence of simulator training within the last 12 months for P1 and 24months for P2. I dont think extra premiums would get you anywhere.
In terms of would you get away with it then the answer is yes .....probably the FAA if they happened to be on the same ramp as you could revoke your license and ground the aircraft but the odds are in yuor favour - your choice
You need to do the CPL & IR exams and 5 hours minimum training for each ride. You then could approach a 135 operator in the US with your new ticket and use their internal training program for the SIC at a cost. Yuo could probably then get it issued at a local FSDO with less questions