Hi Ynaig
I have read, listened to and seen most of Byron Katie's and Abraham's Books, CDS and DVDS, many times, so I chose not to read the links above (in case they influence my perspective) and just try and offer my understanding of how they both work together.
First of all, Byron Katie does not use the word "Desire" like Abraham/LOA does. BUT she does use the word "Preference", which is similar. Every moment, we make choices ... do I go this or that way, do I accept or reject this thought, do I eat this or that right now.
It is impossible not to have desire/preference. And each moment, a preference is there. When we don't have money, we want it, When we have sickness, we want health.
The thing about Byron Katie is that she is clear enough to say she prefers (desires) what is in front of her. She is able to say that because she does not live in a duality world, where there is right or wrong or only judgement of what should or should not be.
No matter what the conditions are, LOA is about feeling good NOW, or conditions can not improve. Byron Katie's work, creates that peace by releasing judgement. From LOA, the only work to be done is to make an effort to find relief from where you are, and relief comes from releasing judgement and stoping trying to control the world around us, so, both really work well together.
Have you done "The Work"? Those who use this technique regularly find that the feeling at the end is peace. Peace in not non thought or non action. Peace releases us to our highest good and often a lot of action takes place. Inspired action, inspired choices, inspired preferences/desire.
When we make judgements and try and control, we give too much energy to the impossible. In doing "The Work" the result is always unconditional Love to self and others. Because any time we feel negative emotion about anyone or anything, we have deviated from Source energy. When we feel peaceful and happy, then we are connected. In connection, that which is our preference/desire, easily manifests.
And you mention "sufficiency" .. that is a great LOA stance. When we are happy with what we have, we attract more of the same. In LOA terms, wanting something we don't have is focusing upstream. We can not get to the place of our inner desires when we focus on lack. But, when we are at peace and feel sufficient with where we are and what we have (vibrational allignment), we are in a state of allowing, non resistence. At that point, we naturally attract all that is in our highest good, all that is meant to be.
Everything that happens to us is a by product of our dominent thoughts. So, even if one is not aware of desire or preferences, they take place and are mirrored by all that we experience in our lives.
Byron Katie's work is rather Zen like, she doesn't subscribe to the eternalness of our being, and that we come into the physical from a variety of perspectives so that we can explore ideas, situations etc. and that is a core element in LOA/Abraham's work.
I think I've deviated and not quite sure if I answered your questions!
Love
Maya