@El Cid
I don't have much experience with the bent press, but I've done some extended periods of PTTP side pressing with an olympic barbell.
IMO, the toughest implement to press is the DB. The KB hangs below the hand and is therefore more stable in the groove. All the weight of the DB is out at the end of the arm, which makes it harder to keep the groove.
The barbell seems like it would be more cumbersome than the DB, but in my experience it is actually more stable. I think of the effect as being like a tightrope walker's pole. The length of the pole adds to the stability of the system.
A key factor in pressing the barbell with one hand is finding the right balance point. Don't center you hand on the bar. Instead, center a point nearer to the base of your pinky finger. In other words, move the thumb side of your hand a little out toward the collar. Once you learn to find the right balance point, the bar will be very stable, and you don't need to grip it tightly to keep it in balance. You can actually press it with an open hand, although a tight grip helps with generating overall tension.
My side press style is right out of PTTP -- not close to a full bent press. It is more like a military press with more lean and more rotation to get the elbow more on the lat, but not nearly as dramatically as a bent press. It's a little like a behind the neck press groove, but with some rotation as well. With this style, I can side press a lot more weight with an olympic bar than I can with a KB. My best KB military press or side press is 36kg (maybe I could press a little more, but I can't press 40kg in either style). My best olympic bar side press by comparison is 110lbs x 1 and 95lbs x 5.
BTW, with the olympic bar, I either take it off a rack at shoulder height or cheat snatch it with two hands to overhead and then lower it to the starting position. I also find that I can get a better, tighter starting position by getting the bar to overhead first (any way possible, by jerking it, snatching it, and/or using the other hand to assist) and lowering it into position.
Hope this helps.