Figure out how much tax was withheld without form 1042-S

Maybe. A wage and income transcript is claimed to cover 'information returns ... such as W-2, 1099, 1098 and 5498 ...' (emphasis added). Those are the information returns usually of interest to a 'domestic' (citizen or resident) taxpayer, but 1042-S definitely is considered an information return (see e.g. pub 1179) so logically it should be included, although I can't say for certain.

According to NTA a while back, only about 1/3 of people who try pass the security checks for IRS online applications -- and presumably most or nearly all of them are residents -- so I'd say your chances of online (thus immediate) access are not great, but it doesn't hurt to try. It appears the online-to-mail option doesn't support this type, so you may have to download form 4506T and mail or fax it in, after which processing and sending paper will probably take a few weeks. Good luck.