I’ve got a Musician Pegasus R2R NOS DAC that, while I wouldn’t call it warm, it is certainly natural and musical sounding with no sense of digititis. It’s got a small footprint and at only $1099 is well within your budget. For about $2k more you can step up to their Aquarius flagship DAC that gives a bit more of everything. Here’s a review, and you can find a review of the Aquarius on the same site.
https://soundnews.net/sources/dacs/musician-audio-pegasus-r2r-dac-review/
Now here’s where it gets even better — you can add their Pisces DDC that re-clocks the data and allows you to take advantage of their DAC’s i2S input via HDMI that significantly boosts the performance even higher (from personal experience), and at $619 you’d still be well within your budget with either DAC above. It shares the same footprint and form factor as the DACs so they look good together, and even with the DDC stacked on top of the DAC they’d probably still be around the same size as the Border Patrol DAC. As with the DACs you could step up to their Leo or flagship Phoenix DDC and get better clocks, etc. and still be within budget.
https://soundnews.net/accessories/level-up-your-dac-musician-phoenix-ddc-review/
Just another option to ponder, and best of luck.