As shown in hissingnoise's link, the use of barium peroxide precipitates the sparingly soluble barium sulfate, leaving a nearly clean
solution of hydrogen peroxide.
I've made barium peroxide by heating barium nitrate to 1 000°C in a SS crucible with an oxy-methane torch:
Ba(NO3)2 → BaO + 2 NO + 1 1/2 O2
Then I cooled it down to 500°C and blew pure oxygen over it for ten minutes while maintaining that temperature.
2 BaO + O2 → 2BaO2
Now I had much better things to do with this barium peroxide then make such an OTC material as dilute hydrogen peroxide, but if it's worth it to you,
this is rather easy.