First, I would stop teaching sight words. You're absolutely right that he is learning to only look at part of the word, which is not something you want to encourage. The words you linked are words that he should be learning to decode. When his decoding has improved, you can decide whether you need to drill true sight words.
Had you used a phonics program before starting Barton? I think it's still too early to tell whether Barton is going to work, but the reading "specialists" may not want to admit that phonics instruction, or at least phonics instruction on its own, does not work for every child. Ds16, who is bright, focused and hard-working, received one-on-one phonics instruction for nearly 5 years and didn't progress past a first grade reading level.
When phonics doesn't work, some specialists jump to "sight word" or "whole word" reading instruction, which should really only be a last resort. But, there are options between teaching phonics and teaching sight words (linguistically). If a child can't learn to read with phonics, it is likely because they have deficits in phonological awareness, especially in hearing and identifying phonemes. To bypass these deficits, I've taught several of my children to read by teaching them to recognize each syllable as an onset and rime, not individual phonemes. So, "cat" isn't "c-a-t", it's "c-at", "track" isn't "t-r-a-ck", it's "tr-ack" and "mankind" isn't "m-a-n/k-i-n-d", it's "m-an/k-ind".
Glass-Analysis is the only program I know of that teaches decoding this way, and I've been able to adapt it (with the help of a psychologist who researches reading and dyslexia) to meet my kids' individual needs. I've used this program both on its own and alongside a phonics program, so I would discuss it with his tutor and see whether it would be worth teaching at home.