Reddit reviews PowerBear LG G5 Battery Case [4000 mAh] Up to 140% More Battery

The G5 sports quick charging and Powerbear makes an extended battery case, which might help your battery situation. And fwiw the G5 customers I've run into say that the battery life is fine, and the GSMArena/PhoneArena tests use whatever software is out at the time (Marshmallow I think). The G5 is now running Nougat and that may have improved battery life.

Also if you need amazing battery life I'd raise the budget by $50 and grab a used Moto Z Play (not the Droid version). It has by far the best battery life of any Android phone and you can get extended battery mods for it to get 16 hours of SoT or whatever ridiculous shit /r/MotoZ users are pulling off. Plus the software is crazy well optimized so it runs nearly as well as the OnePlus 3/3T (though it can't keep as many apps open cause it doesn't have the 6gb of RAM).