Another option would be Shout Wipes that come in individual packets, or Shout Stain stick which is a solid roll on stain remover that works up to 7 days....not the gel form. (I use this at home and it is wonderful esp if you can't get your laundry done immediately)
The TSA issue I think with the Tide Stain stick is that it is a marker-like item which means the liquid is contained in the barrel...not only is the 3 oz rule apply here but as you mentioned the solvent rule would also apply.
People have mentioned that things get by TSA all the time. I am sure they do miss things from time to time...You hear those stories all the time. ....there is a lot to look for on those Xray screens. But the exception would not make it a rule. Bottom line, if you know it might be confiscated then be prepared to part with it.