Oh Rowan - you know that the HV is talking nonsense, don't you? Silent reflux is so hard to get a diagnosis for and as I think I said earlier, a lot of GP's dismiss it, especially this early on. I really feel for you because at 6 weeks you are on your knees anyway with a newborn and the gruelling sleep deprivation.
Where do you live? If you are in London, I can recommend the specialist we saw.
I second what others have said - silent reflux can also be a symptom of CMP intollerence. In fact, when DS2 was discharged from consultant care at about 14 months, he said he would never know what the exact diagnosis was - you just treat until the symptoms are relieved and adjust as they grow older. DS2 probably grew out of an intollerence or his stomach sphincter matured.
Anyway, he ONLY formula on the market which is truly dairy free is Neocate (all the others are made from milk and heavily broken down. Neocate is totally synthetic). The problem is that it costs £30 a tin, so doctors are very slow to prescribe it. I went to Boots and bought a tin over the counter (pretend your baby is already on it and that you have run out etc). From the first bottle he drained it, rather than take a few sips and scream etc. I also found the Dr Brown bottles helpful.
Then go to your doctor and, assuming your LO feeds better, insist that they give you an ongoing prescription.
Additionally there are meds which usually the doctors will try in this order:
Infant Gaviscon (you can buy this over the counter too so at least you can say to the GP it is useless if it is! I found it was!)
Ranitadine - this neutralises stomach acid so that although the baby will still reflux, they don't experience the burn. Sadly this didn't work for us.
Omeprazole - this stops all acid production altogether. It takes about 10 days to work. It was a godsend for us and the only thing that worked in conjunction with Neocate.
I also found that feeding DS2 in a baby bjorn chair really helped. He didn't like being held whilst feeding. Please also try settling him on his tummy - it really helps.
Thinking of you