Hiya Worzel,
I'm afraid I can't help from a mother point of view as am just expecting my first next month! But I'm a neonatal nurse so have to deal with this on a day to day basis (though may just be re-itterating what nurses have told you already?!).
It's certainly possible to fully breastfeed a preemie, but does take more work than feeding a term infant, particularly since most units aim for discharge at 35-36 weeks. Don't let yourself be presurised into giving a bottle unless you want to yourself. If you have enough milk when you express, you should be able to feed and most of the time it just takes patience. I know that seems so obvious, and it must feel like he's been in hospital for ages, but he WILL get there, he just needs to get there in his own time. I hosnestly think that in some respects, this time for parents is the hardest as before when babies are sick they can see a efinete improvement every few days in their babies condition. They come off ventilation, then CPAP, they're tube feeding instead of having TPN, they can take a bit of milk orally, they're transferred into HDU or nursery and put into a cot... etc etc and then all of a sudden the only thing connecting you to the unit is a nasogastric tube!
At almost 34 weeks he sounds like he's right on track for his gestation tbh. Does he have to be 2hourly fed- is he still having resp problems, vomiting etc? If not, could they try 3hourly feeds for him as while he's being fed every 2 his tummy won't have a chance to empty. You will probably notice a big difference in his energy levels after this change. Try putting him to the breast every 6 hours, and leave him sleeping soundly for 'rest time' in between. If his weight gain is good at the end of the weekend/ beginning of next week, ask the nurses if they can give him a reduced feed the time before you put him to the breast to see if that makes a difference. Eg, if you're coming in at 10am to feed and he's usually having 20mls 2 hourly, ask if at his 8am feed they can just give him 10mls and see how he feeds after that. You're right that very often babies are 'over fed' to help with weight gain in the early days and becuase of this they don't get hungry. If he still feeds poorly at the 10am feed, you will know it's just a bit soon for him and can try the reduced feed method again in a few days.
Other than that it really is a waiting game at the moment! Does he latch on and manage to stay latched on, even for a minute or so when he feeds (even if he's not continuously sucking during this time)? Once he's feeding for 5 or 10 mins fro you they can start givng reduced top ups and it is at this point you may need to come in more to establish the feeding, but this may not be for another week or two yet.
I hope this was of some help, have been folowing your story, can't believe he's already 6 weeks old!