With “not so good water bottle pockets”, maybe you need some not-so-good water bottles? i.e. smaller, shorter water bottles. Maybe that’s incompatible with your water treatment scheme that needs a particular screw top or a largish opening for a SteriPen, but otherwise, try a little dumpster diving at the recycling center and try a number of different skills / sizes / heights / widths.
I’m not totally sure of the scale, but the first thing that came to my mind was a somewhat short, squat bottle and that square-cross-section bottles are often more secure in a side pocket. Something like the smaller, shorter Fiji brand bottles:

As for modifying your pack: try a safety pin first to snug up the top of the pocket and see if that gives you enough tension to hold a bottle reliably. If so, it would just be a few stitches to take up some slack. Or to add some elastic (stretched when you sew it on) to the top of the pocket.
Or return it and test other packs in the store with large, full, round water bottles. I certainly like the option of having large water bottles convenient but secure..