Just how much time do you have at SIN? Depending on how much time you have and where you arrive and depart, you might be better off to try to get a shower there. Last time I was in SIN the T2 lounges are almost deserted now that SQ has moved most of their flights to T3.

At BKK there is no problem with respect to immigration if you want to visit the international ROP lounge first. Just make your way up to level 3 after arriving and show them your boarding pass at either of the ROP lounges on Concourse C (don't go to the big one on Concourse D because they don't have any showers). My guess is that they will allow you in being you meet the requirements for entry: a TG J boarding pass for that day. Ask immediately at the desk for a shower. If it's not a busy time they should have one ready for you to use but if at a busy time you may have to wait. After your shower, go back down to Level 2 and pass through the immigration / passport check next to the International-to-Domestic transfer area, which is where Concourse D and C meet. Do not go to the normal immigration. You will then be let into the domestic concourses (A & B) and if you still have time you can then go to the TG domestic ROP lounge on Concourse A.