... SEP rating is about to expire and I have no way of renewing it in Singapore and my FAA license depends on the currency of the CAA license.
I'm not sure that's true. The restricted US pilot certificate is kept valid by complying with the US regulations, under which class ratings do not expire, rather than those governing the base licence. This position was clarified by the Assistant Chief Counsel for Regulations in an interpretation given on 22 Mar 2012 to Andrew Krausz:
Further complicating your inquiry is a notice issued by the Aviation
Accident Investigations Board (AAIB) in August 2008, which quotes the
14 C.F.R. §61.75(e)(3) which states the US certificate, "Is subject to
the limitations and restrictions on the person's US certificate and
foreign pilot license when exercising the privilege of that US pilot
certificate in an aircraft of US registry operating within or outside
the United States." The AAIB draws the conclusion from this provision
that any and all limitations and restrictions that a pilot would be
subject to under his foreign pilot certificate are incorporated in his
US certificate, and apply equally under his US certificate. This
conclusion is mistaken.
While 14 C.F.R. 61.75 does incorporate the limitations and restriction
"on the person's US certificate and foreign pilot license," (emphasis
added) that language refers to the scope of the authority reflected by
the certificate itself. In other words, the pilot is subject to the
restrictions and limitations that appear on the face of the US
certificate or foreign pilot license. The language does not include
the entirety of regulatory requirements of the foreign State since the
holder of the §61.75 certificate is bound by the US regulatory
requirements to exercise the privileges of the US certificate. The FAA
views that language as addressing the limitations of the sort FAA
uses, e.g., "not valid for night operation," where the individual has
not completed the night training requirements.
Having a choice of anywhere to fly in the US, what seems like a good idea?
The proficiency check done to renew an expired UK/EASA class rating will be less onerous than the initial skill test for licence and class rating issue. There are UK/EASA examiners in California (San Diego), Arizona (Phoenix), and Florida (Naples, Melbourne, Sebastian, Jacksonville, Fort Pierce, St Augustine, etc). Among these the best climate at this time of the year is in San Diego (Weatherspark comparison
here).
Otherwise, US training need only be pursued at a Part 141 FAA-approved flight school if entering the US under a student visa. Note that training which leads to a certificate or rating cannot be done under a B-class visa. If you have the the option of receiving training from freelance instructors then you might consult the Society of American Flight Educators (SAFE) directory of instructors at
https://www.safepilots.org/member-pr...safe-educator/ An alternative directory which includes instructors who are National Association of Flight Instructors (NAFI) members is at
https://www.iflightplanner.com/FlightInstructors/ If you intend to carry on with non-commercial flying during business trips to Georgia then consider doing the training there.
Unless you are a US national a TSA security threat assessment must be completed before receiving any flight training towards a US instrument or multiengine rating or initial standard US pilot certificate. Expect the assessment(s) to take anywhere from one to three weeks.