Faa Ppl/ir
Don't know about ME but for SE you need 40 hrs of which 15 can be in a sim and at around £100 an hour average depending upon location you can reckon on $4k to $5k. There are not generally any charges for approaches - yet
I understand you need to uprate the IR if you want to use it with ME so you might be better taking it ME to start with - check. Twin costs in the US in my limited knowledge are from around $170/hr.
Don't know if there are any charges for the paperwork/ VISa yet.
Have you got anywhere in mind ? I was wanting to take my FAA IR SE in the US and was looking for a pilot to buddy up with. PM me for more details if you are interested.