Go back to a and e if no joy at GP on friday. I wouldn't usually advocate doing that but I have been in your situation and did just that. We were desperate, and it did the trick. Tell them how desperate you're feeling, and how you are absolutely certain your baby's in some sort of discomfort.
Meanwhile, you could always try getting the GP to prescribe a PPI like omeprazole/ lansoprazole directly. They can do this, it's just that many don't like to because they are unlicensed for infants IIRC. Ours prescribed Ds's no problem in the interim, while we were waiting for paed referral to come through.
Think the doula is an excellent idea. Go for it, use your savings, and don't worry about that: you need to get through the next few months with your sanity intact.
Milla is right about the risks of having a limited diet. It is much harder when you're already vegetarian. You will have to ensure you get sufficient calories and nutrients, which is a headache you frankly don't need at the moment.
Think lots of hummus, avocados, ginger biscuits (many are dairy free), dark chocolate (Lindt 70 and 85% is milk free IIRC but may contain soya leicithin if you are also avoiding soya). fortified oat, almond, rice or hazelnut milk can provide additional calcium and something to put on your cereals. I would also take a multivit and mineral supplement if you're not already.
Nevertheless, I would still stand by the milk-free suggestion, because specialised formulas are not necessarily the easy alternative. Aside from one brand (and we have tried them ALL in our time) they taste disgusting and therefore getting your baby to accept them can be challenging. There was a point when we had to flavour our son's with fairly hefty amounts of banana Crusha. Not recommended but it was the only way to get any of it down him (I stopped BF fairly early for a variety of reasons with him). Some babies will drink hydrolysed formulas OK, others not, so the palatability issue may not crop up of course.
Staydown formulas are sometimes recommended for babies with reflux where cow's milk allergy is not a factor. Some people swear by this but others find it makes no difference.
Also, PM me your address if you feel like it as I have a swaddle pod or two lying around that you are welcome to have :)