my two cents...
identify what your priority is. if it's to focus on the half marathon until Sept 1, tone down the lifting, definitely don't increase the weight! focus on getting in at least 3 runs per week minimum with at least one longer run slowly building to about 15km. the other 2-3 runs should be a mix of tempo, hills, easy 5km and sprints.
when i ran my last half i followed a crossfit endurance type schedule which worked really well for me. i did weights (including body weight like pushups, pullups) 2-3x week, ran 3x week.
also, you may want to focus on weight bearing exercises that support your running like squats, lunges, planks/sideplanks, pushups, pullups. these all compliment running muscles and add good stability and balance elements.
i usually only consolidated 1 workout/week.
most importantly...have fun with it!!! if possible, get in a 10km race about midway through (assuming you're new to longer distances).