"Feed to sleep" vs burping

Hi all
Can anyone offer some insight on how to make burping work into the feeding routine? In the hospital, we were taught "feed up to 20 mins one side, burp, feed up to 20 mins the other side, burp."  My newborn gets extremely agitated every time we burp him. I feel if I follow this pattern he's really difficult to calm down to eat on the 2nd side, and after I have a fully fed baby I take him from happy to fussy immediately via burping. I  terrified if I don't burp him he will choke. I am also frustrated I am consistently putting my child into a bad mood by burping, & how much this is interfering with nursing & his sleep.
Any constructive insights? Or strategies to reduce choking risk without making my poor little guy mad all the time?! Thanks!