I see what you mean. He does write a disclaimer that his book deals with hypothyroidism only, and not Graves or thyroid cancer. But as I've got it on Kindle, I did a quick search and one thing I thought was quite interesting in regards to what you've described, is one of the functions of healthy gut bacteria is:
'to facilitate the conversion of T4 to T3. When diets are poor and digestion falters, dysbiosis, an overabundance of bad bacteria, occurs crowding out the beneficial bacteria and hampering thyroid function.'
I don't know what your diet or digestion is like, as I'm assuming you must be on appropriate thyroid meds for you, having been under Dr P AND due to not actually having a thyroid anymore, but as I mentioned before, diet and digestion are things I've been addressing in myself too, slowly over the past two years, and each change has seen a small improvement of sorts. I had IBS and already discovered dairy as one of my triggers back in my 20s, but have actually found going gluten free last Jan, has removed so many more of the symptoms I've been plagued with since I was in nappies. I then took probiotics for a year (high strength/multi-strain 50+ billion CFU) after researching about healthy guts. I no longer take probiotics as I've now moved onto eating fermented vegetables everyday, (sauerkraut is ridiculously easy to make), and I only need 1/4 - 1/2 a cup a day to get more than any capsule form could give me. It's like a yummy condiment now.
I've been toying with the idea of Betaine and digestive enzymes, but again as I mentioned early, I know my blood sugar is high (I'm one point away from what's classified as prediabetes) and I've got a fatty liver, both of which can apparently be reversed by diet, so I'm giving that a crack first. And to be honest, its day one of my second attempt to cut starch out of my diet (having cut sugar out at the same time as the probiotics regime) and I'm suffering major withdrawal symptoms - headache, nausea, gurgly tummy, itchy skin, diarrhoea, dry mouth, fatigue but insomnia (its 2am here at the moment). I know I'm not hungry cos I've eaten good meals all day, and even snacks in between, but it just goes to show how powerful any kind of imbalance can be.
Dr K doesn't bang on about it in his book (it was only in the second reading that I caught certain statements here and there), but he said first you need to address gut inflammation/dysbiosis, anaemia and blood sugar imbalances if you're to heal your adrenals and the dysregulation of other hormones, and I guess even in your case, your body's ability to fully utilise your thyroid meds. I've just started his second book 'Why isn't my brain working?' and its the same starting point there for healing too.
I really like everything these Functional Medicine trained doctors say, and they're all singing from practically the same book. Interesting that them, along with Dr Lee (writes about progesterone deficiency which I'm also trying to correct), say that even polycystic ovaries (have this too) is actually an indication of metabolism problems (caused by a diet high in sugar/starch/refined food).
Another thought might be SIBO or h.pylori or candida overgrowth. I'm guessing candida die off is what's driving my withdrawal symptoms at the moment, as yeast is a problem for me too. The list just goes on and on... LOL
Long story short, but not really - haha, perhaps start researching gut health and metabolism instead and what you could do to restore it naturally. Hope I haven't waffled too much.