Hugs, we have a reflux baby here too. It presents a lot of challenges in so many ways.
Why did they suggest the side lying position to you? Has your lo started on any meds?
TBH, we went through so many feeding issues with Kate, and found her most receptive to feeding while she was sleepy, whether just waking up or going to sleep, so I fed her then, The most important thing to me is that she eats. With my older son (non refluxer), we fed to sleep all the time. He loved lying down and snuggling in like your lo.
Is your lo on meds right now?
I can't give much advice on the keeping awake. I have found myself very sleepy feeding Kte on may occasions, and have dropped off to sleep in the chair with her many days, only to have my 3 year old wake me up!