Sometimes, when I'm drinking Lugols, I hold it in my mouth for a long as I can. This was initially to hopefully deal with any possible mouth/gum infections.
How ever, I also wondered if it might also act "sub-lingually and against the cheeks, providing faster absorption. Of course, only a fraction would get absorbed that way.
Probbaly not what you want from what you have described, but I thought it was interesting to try an alternative absorption path.
Just recently I've had a cold and sore upper/roof of mouth, so I've been gargling with it a lot. Whether it was that or not, but the roof of my mouth feels better now. Still have some cold symptoms though.