I've carried a stainless steel "thermos" filled with hot coffee with me for years. I use the 1 qt. size...small enough to handle and fits nicely into a small pack.
When it comes to the after effects... well, I've worked in healthcare for years and usually manage to get away from the workplace with one of those cheap, disposable, plastic urinals that have sealing lids on them. I leave it stuck upside down on a limb a good distance from my stand, pick it up on the way in, empty it and stick it back on the limb on the way out. Never seems to bother the deer and lasts all season.