I have much the same gear as Michael, only with the P45+. I too have had vibration problems with the 300 mm APO lens.
Here are my findings and partial solutions.
MLU is fine, but even waiting 5-10 seconds doesn't always help. The reason, I've concluded, is the focal plane shutter. Every action has an equal and opposite reaction, and it's that reaction that causes blurring. (I confirmed this by attaching a small mirror to the lens and watching the reflection of a flashlight beam over many feet in a dark room as I operated the shutter with MLU - the rig amplifies any motion.)
Holding the camera on a rock-solid brick wall solves the problem, but the brick wall is very hard to carry around. But no question, the more inertia in one's tripod, the better.
I found the problem worse when using the lens mounted to the tripod, which is logical. It balances the camera equally fore and aft, which more readily permits a see-saw movement when the shutter moves. By losing the tripod lens mount and using the camera mounted directly on the tripod, then there is a lot of mass at the far end of the lens which is harder to lever into movement. (Overall, the 300 is a very light lens - I'd rather it were heavier! I may try draping a sandbag over the lens.)
Shutter speeds from 1/100th and faster are generally fine. This is less of a problem than you might think because the lens is optimally sharp at full aperture anyway. Speeds longer than 1 second look good too. One needs to avoid 1/2 to 1/1/60th sec. A pain! In practice. I tend to up the ISO and try for at least 1/250th sec.
I do not have this problem with any other lens. I think the 75-150 has so much mass it keeps things immobile.
I hope this may help others and I'd be glad to hear others' recommendations.
Bill