re: 49-ways and corners
Nope, no notches. The restriction is perfectly round. I have heard differing opinions on whether or not this is good for fighters, and I have no opinion either way, as I don't play fighters. I do know that a lot of the hardcore fighter players are moving to Sanwa, Semitsu, and the like.
re: Why the P-360 is bad for 4-way games
Well, first, there's no 4-way mode for a P360. You have to change the actuator on the bottom of the stick to change modes. That's a pain if you plan to do it often. But the big problem is the round restriction coupled with the long throw.
A true 4-way stick has a diamond shaped restrictor hole. The points of the diamond are oriented toward the cardinal directions: Up, Down, Left, and Right. The flat sides of the diamond face the diagonals. This does two things: The stick is naturally guided towards the cardinal directions, and it is prevented from going to the diagonals.
A joystick with a round or octagonal restrictor does not do this- the stick will just as happily go to the corners as it will go to the cardinal directions. A stick with a square restrictor will tend to favor the corners over the cardinals.
With a P360, if you don't change the actuator, it is easy to hit diagonals when you don't want to. And hitting diagonals when you're playing a 4-way game can be disastrous. On some games, the on-screen character will choose a direction at random. In others, it will continue in the direction it was last pointed. On some other games, the character will simply stop until you push a cardinal direction again.
If you do change the actuator, you may find that there are large dead spots in the corners, and the stick won't guide itself to the cardinals. Many players, myself included, miss the feel that comes with the diamond restriction and believe it slows down your reaction time.
That's why so many panels still incorporate a dedicated 4-way stick, along with the 8-ways. Or have swapable/modular/rotating panels that allow a 4-way to be swapped in when it's called for. I don't know how possible this is for your coctail setup, though, which is why I suggested the J-stik. It switches between 4 and 8-way modes by just rotating the restrictor plate, which is much easier than an actuator swap, and provides the true 4-way feel. They are available with a red balltop, and have a longer throw than the Mags.
The 49-ways are a trade-off: It's not the best 4-way, but it's better than an 8-way with an actuator swap. You don't get the true 4-way restriction, but the electronic 4-way mode keeps you from accidentally hitting diagonals, and doesn't leave huge dead spots in the corners. It has the longer throw you want, and has the bonus of making 49-way, Hall-effect, and some analog joystick games playable.