The good news - you can do this, and it is worth doing!
The bad news - it may be a process, not something that happens overnight.
My little guy was a preemie, who was losing weight, after being born by c-section. We started bottle feeding him really early on - then working on getting him to nurse. He was not a fan of nursing at first - the bottle was easier. We managed to set up a supplemental system where I would get him close, then put a small tube (connected to a supply of milk) in his mouth along side my nipple, so that he would be getting something when he nursed. When I did that, then he would actually latch and try nursing. (Doing this required 2 people, as you run out of hands to make it work.)
He got the idea that nursing was a good idea. However, it took some time for him to get proficient - and we were told to nurse for ~20 minutes - and then switch back to a bottle to finish out the feeding. We did that for some time before he would be satisfied with 'just nursing'.
My suggestion is to keep offering. See if the pediatrician is OK with you offering at every feeding, but the finishing up with a bottle, if he seems still hungry. (I know this is a ROYAL pain, as you will need to be pumping to make it work... )
Also, work with the LC to see if you have the latch right- as it sounds like you have the milk, but little guy is missing the technique - or energy to make it work.
Try not to stress about this. (Easier said than done, I KNOW.) For me, the realization that I needed to do this for longer than a few days was hard on me. (My first 2 kids did fine, needing no help. The one with the problems was the 3rd - so all my practice did not do me much good.) My little guy got the hang of it, and now at 8 months old we do not need bottles (except for when he and I are separated). (That was true starting at about 6 weeks - but it was 6 weeks of nursing + bottles for each feeding. :( The hardest part of that 6 weeks was not knowing if he would ever get the hang of it... Looking back, it seems much more trivial than it was during the time that we were having problems.)
So, if you are still reading... the important parts.
You can do it.
It will get better.
You are doing a good job - his weight is caught up.
Take care of you, and try not to worry.
Keep offering, and work with the LC if possible.
Good luck, I'll be thinking about you. The first few weeks are the hardest - sleep deprivation is a killer. You sound like you are doing a great job!