This is totally not how to treat a syncope patient (fainting by poor brain oxigenation/blood perfusion). First of all syncopes resolve in seconds or minutes are max with the excpetion of some arrythmia triggered ones (and these are overall medical emergencies) albeit if there were pression related triggers, it could be still altered a bit for some more minutes. A way to speedup the full return of cosciousness is to rise their legs. In a RL scenarios of orthostatic (mechanism that rise blood pressure when rising ) or vasovagal (vagal stimulous) related syncopes (mild heat related one could be one of the 2) in the time of just putting back an underwear most recovers. No princess carrying someone to their room wth their head above the body ( when someone faint you should keep their head lower then the body, and actually you shouldn't move them if there is no immediate risk, you should just lay them to the foor and do the leg rise manouver).
Temperature measurments is fine if you suspect an heat reated illness (heat hexaustion or heat stroke) or an underlying fever trigger , but most of worldwide regulations for hot tubs and baths require a maximum temperature that is safe in this regard.
I'm not bothered by the fainting trope itself, but authors, you should use it in a way more attinent to reality.