Doesn't your PCB EDA allow you to change the paste mask openings per pad? Just order the stencil with the paste file.
This is to say, reduce the paste mask opening for the BGA part; follow the recommended paste mask definition given in the datasheet, or, reduce it even a bit more if you fear shorts.
Then, about soldering, I get good results combining heating from the bottom, PLUS hot air from above. This is to "simulate" a proper reflow oven, which circulates the air both under and over the PCB.
So, use the frying pan (I use a 3D printer headbed at 150 degC, working great) to bring the board close to the reflow temperature, slowly, but not quite there. Make the hot air do the magic. Frying pan takes care of heating the board, while the hot air heats the components (and the board, of course, as well - but it doesn't need to do all of the job of heating the board).