Your LO sounds so much like my son. My son gulped down so much formula! At 3 days old he was sucking down 4 oz at a go.
I thought that was weird (b/c I'd read BW) but my ped said "babies know how much they need to eat." Um, yeah not that much. needless to say a week later he was in awful pain from not being able to digest all that food and the reflux was just excrutiating. Just like you say, they are sucking to soothe the burning, not because they are hungry necessarily.
My son hardly ever cried from hunger either! He'd eat when I fed him, but he wasn't one of those kids that would wake crying to eat. I'm in the U.S., so our medical system has a different way of approaching reflux. If it's silent reflux it is very hard to tell. There definitely are tests -- upper GI test, scoping -- but some docs just elect to medicate first and see what happens.
Have no experience with Gaviscon. If it's silent reflux, there's a chance that it won't be enough medicine to help. But hopefully you will see a difference. We used Prevacid, which was a lifesaver for us. Prevacid takes 2 weeks to start working, and that's about how long it took for us to notice.
Anyway, definitely follow up with the reflux before doing a ton of sleep training. The reflux board has a lot of great ideas on how to reduce symptoms -- e.g. keep LO upright for at least 20 minutes after eating, elevate head of their crib, etc. Go on over there and do some of those things as well adn see if they don't help.
Good luck!