OP, I thought I'd add to the already helpful (although admittedly conflicting!) advice you've had. I've been where you have twice - my first just wouldn't bastard sleep then DC2 has (had?!) silent reflux and possible CMPA and still rarely sleeps through (18 months).
By 9 months old with DC1 I was on the very brink, I don't think I had PND either but by god I was unhinged - deranged. There was other bad stuff going on at home as well , but I was being woken every 45 minutes by a screaming baby who only napped in the day if I'm the buggy and MOVING. No rest during nap times for me, I had to be out pushing the shitting buggy in the worst of the winter wind and rain.
The thought of cosleeping would have sent be over the edge, I needed space, not more time with the baby. I couldn't wait any longer, so the idea of "this will pass" was utterly unbearable, I had NO reserves left. So we sleep trained at 10 months. We did controlled crying which is NOT the same as CIO. It took three days and DC cried significantly and I mean significantly, less on the nights we CC than they did on an ordinary night waking every sleep cycle, so make of that what you will - and has slept through ever since barring illness etc. But importantly, there are even gentler methods and sleep training does not have to mean being harsh on your baby.
BUT of course if reflux or allergies are the issue then this won't help (yet). DC2 didn't respond to ranitidine- there is also the option of omeprazole. Your Paediatrician sounds unhelpful to say the least. It's hard but I think you go back and get demanding. It may help if you can get DF formula. Giving formula does not necessarily mean the end of the road for BF, I successfully combi fed both of mine - but of course you would have to maintain your dietary exclusions. But you could phase in F and see how you and baby feel about it. Please don't feel it has to be all or nothing or even an immediate decision- that's not pressure you need.
Lastly, do you know anyone IRL you can offload to? Most people I know second time around have been much more honest about how shit things can be with sleep and diet and it makes me feel so much more normal! I also have met other Mums with allergy babies which means you can share the struggle. It's damned isolating being with a small baby and it's easy to imagine everyone else is doing better than you.
And don't feel too awful about the cat - I am a proper "cat lady", avid cat lover who thinks animal cruelty is unforgivable but Jesus wept my noisy mogs have come VERY close to getting a wallop after going and shouting outside DCs door when I've just got them to sleep and I'm desperate.
And don't ever think you're failing because of how your baby is - it's luck of the draw, not your parenting skills!