I'd probably abandon the whole PoE part. I would probably want to do 10-baseT over cat-3 with a separate line for power. The cat-3 is rated for 300m/960', which covers 4 out of 5 within spec. So it would be a gamble whether that 1100' run would work or cause issues. If you assume there is a 15% under-rating, then 1100' should work. I'll let the others chime in as I'm just purely guessing on that number
As far as the separate run for power goes, I was just thinking something like 16 or 14 gauge low-voltage cable. In-wall speaker wire comes to my mind. Something that would have minimal loss within the camera's range. And if that's a problem, just get a bigger voltage supply and put a regulator on at the camera end.
Maybe not the most plug-n-play solution, but still not overly complicated or home-brew. Two cables (or siamese) per pole, a switch that you can port limit to 10-baseT, and a PSU.
At least that's my take on your situation.
