Well, unfortunately, adding capacitance to the gates didn't really help. It dampened the oscillations a bit and reduced the rise-times, but didn't do much else.
Reducing Vin to 19V did quite a bit more, but there's still oscillations when the chip switches.
So far it seems as if the problems are primarily related to low power-conditions, when the output-current is less than 1/20 programmed current (I currently have the circuit configured to charge with a maximum current of 10A). When I demand currents in excess of 1.5A, the circuit behaves more or less like it should. Below that, it starts oscillating everywhere, as the control-loops start to do stuff they aren't supposed to. At the same time, the chip also gets almost too hot to touch.
One other thing I noticed: Once I connect a battery, the voltage at the Vfb-Pin starts rising to about 2,7 to 3V, despite the fact that it is configured in a way that 14.40Vbat = 2.465V at the Vfb-Pin.
Somehow there's current flowing out of the Vfb-Pin if the chip isn't doing anything, which in turn prevents it from doing anything because the VA-Amplifier inside the chip effectively disables the chip.
The only source for current coming out of the Vfb-Pin seems to be Buffer-Amp A11, which is connected to the 3CELL-Pin and Vfb-Pin internally via resistors.
Btw. the datasheet explicitly states that the 3CELL-Pin should be left floating if not used.
Also interesting to note: If the load at the output is high enough but not too high to trigger any off-switch condition (Undervolt, Overcurrent, Overvolt at the output), the chip turns into a linear regulator, turning TGATE on for a prolonged period, only occasionally turning it off for a brief moment to recharge BOOST. Because BOOST can be held up for so long, my theory about A2 causing these oscillations is pretty much busted and I have to look at some other component...