B03S is indeed a successor to B01S and is interchangeable with it.
What you're describing is a brake that isn't retracting its pistons correctly.
Air contamination isn't a direct cause of this problem and bleeding itself isn't a solution. However, this problem often occurs on Shimano brakes that have been heavily used with the pistons running very extended (potentially overextended) for a long time. If that's true, it's common for the fluid that comes out of such brakes to be very dirty and good to replace in its own right.
To attempt fixing it in a way that's guaranteed not to hurt it, you do need Shimano mineral oil. Other brake mineral oils sold as generic service replacements (Finish Line, Muc-off, etc) would also be okay.
Take the pads back out and using the brake lever, intentionally extend the pistons such that you can see about as much as you ever would in use. Thoroughly clean the entire brake with something appropriate that won't attack the seals, so either alcohol or a soap/detergent sanctioned for the purpose. Once it's clean, use a cotton swab to slather the exposed pistons with oil, then push them back in. Extend them out again and repeat the process. You're looking to reduce the feeling of friction as you push them in. Once you've done that until you see no further improvement, you could bleed it at that juncture if you wanted to know you were "doing everything." Then clean the caliper again to get all the oil off and leave it with zero residue, then set the brake up again.
Those steps are about as far as you can go with a Shimano brake, and if it doesn't work then the next plan is replace. Aftermarket repair pistons do exist that purport to be able to fix such problems, but a whole new caliper is a better option at that point until you're talking about the more expensive brakes.
