If you have to pump the brakes to get a firm pedal, you might want to recalibrate what a "reputable" shop is. They might be honorable and pretty swell folk, but not have the right technical skills for thorough brake work.
Is this a problem where the pedal goes too far to the floor but is firm when you meet resistance, or is it that the pedal is soft as you meet resistance?
If, after a brake bleed, is the pedal firm and are the brakes functioning properly? If so, then how long driving does it take for the pedal to go soft again and brake performance to decline? If they start firm and gradually get mushy, then the problem is most likely in the rotors pulsing the caliper pistons and inducing air. It could also be that you don't have the residual pressure valves to keep a tiny pressure on the pistons to keep them from retracting too far into the caliper.
If the pedal just never gets firm in the bleeding process, then you should probably do a bench bleed on the master cylinder. Getting a good bleed on the master can sometimes be difficult, so you have to take nothing for granted. Get a small bleeding kit for the master. You can do either a flush bleed or a block bleed. You might have to tilt the master around to dislodge trapped air. If that doesn't firm up the pedal, I would also recommend a brake pressure gauge. There are a variety of them out there, but I believe Speedway has them for around $40.