I dont 'do' side blown - End blown for me , Whistles and Jazz reeds !
I had a problem with the Top Register on clarinet , so I went back to the guy I bought mine from and he said "Show me!" Straight into top Register , NO problem ! Just a case of practice - I think the same will go for getting the embouchure sorted for flute !
There are a few Flautists on The cat , so you may well get some input from them soon !