I could believe that with the issues you mention your detox systems get overwhelmed easily. One way I think of adding more Iodine as being helpful is that you can get into a stuck place with a patch of bromine overwhelming the meager Iodine troops. You'd have just enough Iodine to irritate the bromine, but not enough to really roust it from wherever it's being disturbed.
If you're using Lugol's, you could try putting it into a Vitamin C solution or orange juice. Sometimes the iodine part is not easily tolerated, but taking all Iodide works more smoothly. This is what OJ or Vit C would accomplish.
Generally the worse the initial reaction, the more the iodine/iodide is needed. You may be a candidate for using a higher dose once, then backing off to allow the detox to process, then hitting the iodine again -- sort of a seesaw that allows the dose to go higher and higher with detox intervals.
You are your own navigator, so proceed as it makes sense. A minimum goal of Lugol's is 100 mg or SSKI 130 mg.