Daniel,
I have hunted all three and will give my opinion for what it is worth.
I have hunted rutting elk in Arizona. If I were you, I would start putting in for limited areas for elk out west and wait a while to hunt them. You might find an OTC or general tag that could compare, but I really doubt it and if so, it will take a TON of research and many, many years of trial and error to find it. IMO waiting for a decent LE units puts you in line for a GREAT experience.
I hunted the Mulchatna herd in Alaska before the big decline. Caribou are cool, but they are real hit and miss anywhere you go (except the NWT-McKay Lake but that is really not DIY). There are 100 horror stories for every good one when it comes to hunting caribou in Quebec; if someone GAVE me a hunt in Quebec, I'd decline it.
I have hunted Kodiak Island in November during the Sitka Blacktail rut. If I was forced to pick one and only one animal to hunt each year for the rest of my life, it would be the Sitka Blacktail on Kodiak. It is the most beautiful place I have ever been and I have been all over from the bottom of Texas to the northern tip of the furthest northern hunting concession in the Yukon and quite a few spectacular places in between. The double throat patch and thick black forehead just do something for me. The meat is awesome on the Sitkas as well; especially the liver (and I am not much of a liver guy).
With that said, I don't know if I could ever camp on Kodiak...those big bears intimidate me when I just have my bow. So, I am going this fall to POW to hunt blacktails and black bears--they don't scare me so much
It is the easist hunt to do with a group too. I'd look into a boat trip on Kodiak in Nov. The deer are not so high like they are early, you eat like kings, and you don't have to sleep with the bears.