I know this is lower than your budget and its not amoled but moto g60/g40 fusion might be worth considering because it has 6000mAh battery and the display is not that worse compared to amoled in this range. Also has 120hz refresh rate. plus 6.8" its the biggest display you’ll find these days.
gifted Moto G60 to my parents a week ago…device works like a charm for normal day users with once a day charge requirement (since my mother is a bit of youtube fan these days) but my father has rejected the device for being too large to fit in the pocket. So if we are comparing sizes its always 40% outside of your shirt pocket and shines like a brick in your pants.
PS: i didnt dared to put it into my back pocket and sit somewhere to check the results.