I'm guessing something wrong mechanically with the Bb (or its mouthpiece). Could be leaky spit valves, foreign objects or whatever. Try swapping instruments with a friend to find out if it is the trumpet or not.
As for exercises, long tones are good, as are lip slurs, and the magic bubble concept can help.
Time for one of the more esoteric Vulgano Brother observations. When we play a note, the air column inside the instrument has defined and mathematically predictable areas of high pressure and no pressure. In physics these are known as nodes and anti-nodes. The higher the tone, the more of these nodes instide the instrument. With a horn of sufficient light weight, we can play a long tone we can gently run a finger around the leadpipe and/or bell and feel some of the vibrations. Change to a different harmonic and that place will move.
Now for the esoteric part. Playing a long tone, we can shift our awareness to inside the trumpet, and imagine/feel a point of resistance somewhere inside the horn. I call these "magic bubbles." To slur up, we can "blow" this magic bubble further away, backing off will allow the magic bubble to return to its place closer to the mouthpiece.
Our body will memorize the feel of these notes and nodes much more quickly than the cognitive control of several variables can. Remember that the embouchure is (or should be, in the Zen Vulgano philosophy) formed in part by the note that it is playing.
Experiment a bit, and have fun!