dmesg | grep -i i2c shows
[ 1.027247] omap_i2c 4802a000.i2c: bus 1 rev0.11 at 1000 kHz
My test program shows a net rate of around 100000 bytes per second, which seems roughly right. According to my scope, the clock frequency is 909 KHz. The difference could be some kind of rounding error, or granularity in timing? (Ie, exactly 1000 KHz might be impossible to configure with this hardware, and 909KHz is closest.)
The AM335x Technical Reference Manual seems a bit vague. It says 100 KHz and 400 KHz are supported, but nothing explicit about the maximum clock speed. It does say
"High Speed (3.4 MBPS) operation Not supported"
The I2C spec contains "standard mode" (100KHz), "fast mode" (400KHz) and "High-speed mode" (with "up to" 3.4 Mbit/s). Maybe the AM335x controller can't go quite as fast, and technically there's no name for a mode with a ~1MHz clock, so that's why they don't mention it.
In any case, I left the transfers running for a couple hours now and everything seems stable.