$800-$1500 Integrated amps with a subwoofer output

What do you mean by a sub. output ? Do you mean an output that passes only low frequencies ? I've never heard of this though I guess it could exist. Is it for music or HT ?

Some subs (REL) are connected to the speaker terminals, so all integrated amps would qualify. Some subs only connect to pre-out (RCA or XLR) so you'd need an integrated amp with pre-out (which most have these days).

Connecting a sub in parallel to the speaker terminals had absolutely no effect on the sound quality of the main speakers for me. The signal to the main speakers doesn't go through the subwoofer ... it is passed unfiltered.

I use Densen B100,REL strata3, Green Mountain Europas.
The sound is fantastic.

Unless you already own a sub that only accepts line level inputs I think you are unnecessarily restricting yourself.

Suits-me makes a very good observation. Running a long set of interconnects from a passive preamp can audibly affect the frequency response (due to cable capacitance), so it would be better to get an integrated with an active preamp if you want to use RCA interconnects to drive the sub. Unfortunately, since the demise of phono inputs many integrateds have a passive-pre.
Or just get a REL / ACI sub and connect to the speaker terminals.