I've done quite a few physicals that required a drug screen also. I was always allowed to redress before the UA.
Pickup a couple extra of the cheap hard warmers. You can get them a dollar store.
My system:
I nuke the QF for 10 seconds and then crotch it.
When I pull into the lab I transfer it to a pocket with the hand warmer. Do physical. Redress and put QF back in my crotch.
Do the UA. They usually have you empty your pockets.
I usually keep a digital thermometer in my sock to double check temps. Don't get one with the alarm in it.lol
If by chance when you pour the QF in the cup and its a little high just blow on it. If its a little low rub your thumb against your pants and then hold it against the temp strip on the cup for a few seconds. Its all very easy just practice your technique for keeping the temp right.