I wouldn't use toothpaste. That could actually be damaging. The whole point of toothpaste is that it has hard, tiny bits in it to scrape stuff off of our teeth. Basically, rubbing it on any sensitive skin would result in hundreds of little cuts that would sting like hell until the skin has healed.
This may depend upon the brand of toothpaste. I’ve used toothpaste in the past without such issues but different brands may have different effects and different people’s skin will also respond differently.
Best to test on a small less-sensitive area first, if OK test a small amount on a more sensitive area etc and be sure the effect is appropriate.
Alternatively, you can get peppermint oil diluted for bdsm purposes, but that doesn’t stick to the household item requirement.
Cut pieces of ginger might also work but again best to test the skin carefully first.