Hi Worried,
I am sorry that you are struggling to get help. Based on my personal experience, I do not think that reflux is out of the question.
My son was also very happy on his back (though sadly he was not - and still is not! - an excellent sleeper) - he was happy to lie on his back at night and to play on his back during the day (though he screamed in the pram).
His reflux only started when he was about 16 or 17 weeks - before that, he had fed very happily - and though it did eventually get better it definitely got worse for a few months before it improved.
Luckily, our consultant (we had already been referred in his early weeks due to his poor weight gain, so were already being seen regularly) diagnosed reflux straight away. I was a bit doubtful at first (for all the reasons above) but I saw a difference as soon as he started on the medication.
I guess the real question is how the doctors you have seen feel about the slowing weight gain. If they are not worried (or not worried for now), then you probably don't need to push for anything more (though keep monitoring it closely, obviously). But if they are worried, and are going to push you to give him top-up feeds (which should not be necessary at this point if everything was normal, because breastfeeding should be very well established by now) then push back about the possibility of reflux. However, since you have only found out about the poor weight gain today, it is probably best to go back to the doctor / see a different one and present the weight gain and your other concerns.
Can you find a consultant who specialises in infant feeding? As I said, we were lucky in a way because the consultant that we were already seeing (because of poor weight gain) was a specialist in infant feeding. I had always though that reflux could only ever get better, not worse, but our consultant told us that many of the patients that she has seen have only developed reflux about when our little boy did - between 3 and 4 months old.
Let me know if I can offer any more info / help - I know how stressful it is when you feel that baby is not gaining weight at the right rate! My boy gained almost no weight during his first 4 weeks - he only regained his birthweight when he was about 5 weeks old.