Just wanted to sympathize, I'm afraid I don't really have a solution for you!
Both my kids liked to be bounced and walked to sleep at this age. DS2 had reflux so, added bonus, he would vomit on me too!
My first baby didnt have reflux, honestly the difference between my two babies' sleep was entirely driven by sodding reflux. Poor DS2, he was in so much discomfort, it made both our lives a misery really!
I got a few slings, found them absolutely marvellous (not at bedtime, but great for naps). I could get DS to fall asleep in the sling and then sort of lower myself onto a heap of cushions on the sofa, and rest for a bit.
My only other advice is, if he is getting overtired, maybe just squeeze up the entire sleep/eat/play cycle and anticipate the nap a bit sooner? Maybe he is just too tired to suckle (given that you will have to burp him, and perhaps it makes his tummy hurt or feel squirmy?) So he is preferring to be soothed to sleep instead. In some ways although it is exhausting it is better to avoid feeding to sleep, so you can console yourself with that thought at least.
PS with both my babies sleep - feed - play - sleep was definitely the best way to organise things. Sleep - play - eat - sleep was no good as you need time to burp, do bicycle legs, get over the vomiting (in my DS case), change, etc.
I'm rambling. Hope some of this might help.