What you said about him not feeding unless you caught him before he was fully awake - that reminded me so much of my DD.
She was extremely hard to feed. She was exclusively breastfed, would never take a bottle (of anything!), and would feed for a few seconds, or a minute or two, then pull off and get distracted. If I picked her up to feed her when she was not fully awake, I could get her latched on before she really realised (!), and she fed a bit better. She was my second BF baby and completely different from my DS.
She was 9 lbs at birth, but put on little weight and the HV was concerned, though she couldn't offer much advice. She said she had not come across this before
.
I found this all incredibly stressful. I tried switching to FF in the hope that she would take a feed from someone else, but she refused everything. In the end, I persisted with BFing, and she did continue to put on little bits of weight. However, she slept through the night at about 10 weeks.
I weaned her as soon as possible - once she was taking solid food things improved - she put on weight, and the pressure was off me IYSWIM. I BF her at least occasionally till she was 12 months, she never took a bottle, and went straight on to an open cup.
However, this was all a good number of years ago - my DD is now a strapping 18 year old! - and I was 'allowed' to wean her at 4 months, so perhaps not as difficult as trying to persist to 6 months. But I clearly remember my frustration!
I would say don't be too hard on yourself - you are offering feeds, and some milk is getting into the baby! Some babies are just not keen feeders, but are interested in everything else around them! And if he is healthy and alert, I think it may just be part of his character. And, IME, it does get better.