Another 2082X and I also don't use the touchscreen. The unit's been great so far, I've had it running continuously for a few days now trying to test the accuracy of a circuit and a frequency counter, no problems so far. I was also interested in audio band signal generation and from my brief research, the 2KX series was the only ones with such low harmonic distortion. You have pretty complete control over harmonics, but I don't know if you can stack several up (you can pick phase, amplitude, harmonic order, etc, but you can't have a 2nd, 4th, and 6th harmonic built into the same signal). Maybe I've just missed it - though if you have a program to generated that on your computer, just make it into an arb waveform and get it on the unit, the sample/speed limitations on arb functions for any of the 2KX series is plenty fast enough for audio work.
As for triggering for a sweep, if you've got two channels that can be linked it's no problem. Square wave of whatever duty cycle (<100%) that has the same period as the duration of your sweep. Trigger off the square, input off the sweep.
I assume the 1kX series is brand new? It wasn't around when I was in the market just a couple of months ago. I'd be curious if there's much performance difference in the lower frequencies, the front plate and UI look identical. Seems like siglent's datasheet link is dead at the moment, so I can't tell for sure.