I thought the beads were useful to separate larger fragments from small MW contaminants and primers etc. But not so useful to size separate in that limited size range.
In my opinion, to achieve a double size selection you should need two consecutive selections: the first one with 'higher' PEG, so you precipitate fragments < 2Kb, and the second one with 'lower' PEG, so you precipitate fragments >1.5 Kb. Whether size selection can be so sensitive I'm not sure, and I would expect substantially low efficiencies when doing two steps.
If you can't achieve the goal with the beads with your expected efficiency, I would do gel extraction and excise the desired range, and then purify with a method of choice.