$800-$1500 Integrated amps with a subwoofer output

I believe the Manley Stingray integrated can be ordered with a sub out. It's a nice amp and can be found used at your price range.

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.

The Blue Circle integrated can be ordered with a $75 sub out as an option, I believe. It has a robust transformer and pretty good level of capacitance for its price. Point to point wired.

When looking at integrateds to use with a sub via rca out you need to know if the preamp inside is passive. That can complicate hooking up a sub this way, but may not rule it out. I know the Creek and the Jolida tube integrateds have passive preamps.

Agree with seandtaylor99. I currently run a REL Strata with my Sonus Fabers using a VAC tube integrated amp with great results.

Sounds like you are looking for a specific subwoofer output that usually comes with a receiver for home theater. For two channel, there are similar threads out to look at different hook ups. Personally, I don't think a specific sub out is necessary.

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.