It's just really firmly stuck. What would an LBS do?
Use other 'specialist' tools. Bench vices and whatnot.
Assuming the tool is the right one:
My 'pro' tip: bolt the tool on, if possible, and, with the wheels on the bike lean it against something where it's not going to fall over. Attach the wrench parallel to the ground, ensuring pushing down will turn in the correct way to loosen the BB. Then stand on the wrench with one foot, and use your weight to dislodge the BB. Steady the bike with one hand on the seat, the other on the front brake.
Double triple check you are turning it the right way.
Use other 'specialist' tools. Bench vices and whatnot.
Assuming the tool is the right one:
My 'pro' tip: bolt the tool on, if possible, and, with the wheels on the bike lean it against something where it's not going to fall over. Attach the wrench parallel to the ground, ensuring pushing down will turn in the correct way to loosen the BB. Then stand on the wrench with one foot, and use your weight to dislodge the BB. Steady the bike with one hand on the seat, the other on the front brake.
Double triple check you are turning it the right way.