UA & SQ should as *A carriers be able to check thru. Even if they refuse, the transit hotel is a viable alternative. It is before immigration and after your stay just go either to the transit desk and ask for the bag to be transferred, or go thru immigration and pick up your waiting bag at lost and found. Do not exit customs because your bag is in the customs area. Once you have your bag, exit and go to the appropriate terminal for check-in.

Transit hotel response has improved recently so send them an e-mail with arrival times. Remember the actual date you will be arriving for your reservation.

Otherwise, if you do not want to use the transit hotel which is IMO perfectly fine, then the other option is the Crowne Plaza adjacent to Terminal 3.