12 ga....
When you say "hunt for no more than 3 days" do you understand that you can't fly in and hunt the same day in Alaska? So, to hunt 3 days (actually 2 1/2 to allow for flying out on the 3rd day) you really need 4 days for your hunting trip. Otherwise, you'll be hunting only 1 1/2 days if you are flying out and back to Anchorage on your last hunting day. This is something very important to consider when you're making your plans.
OK, where to go. There's only one place I know of where you can meet all your criteria, including keeping your costs reasonable and having a good chance of a successful hunt. The Alaska Peninsula. You can fly commercially from Anchorage to King Salmon direct. Then the same day fly down the AK Pen. with either Pen. Air in KS or King Air in Naknek (12 miles to Naknek on the only paved road. 12 miles is all there is!) (There may be other air taxi services there now, so check it out.) Since it's your charter, you are flying until you find the large peninsula caribou herd south of KS. They will be in bunches and all moving north. Your charter will land you ahead of the herd on one of the many sand blows or on a river gravel bar. The pilot will pick you up mid day on your last day and fly you to King Salmon for your evening flight back to Anchorage. So, that is either a day and a half hunt or a 2 1/2 day hunt depending on whether you have 3 days PLUS a "no hunting day" to fly in, or 3 days INCLUDING the no hunting day flying in. (Don't even think about shooting a caribou even if it walks right up to your tent that first day. If you're caught you will pay dearly, including a large fine, loss of your rifle, and loss of AK hunting privileges. Also, there was a time as I recall when ADF&G allowed same day hunting for caribou on the AK Pen because the herd had grown so big. Check that out.)
The advantage of hunting out of KS is that it offers you direct commercial transportation from Anchorage, bush plane service when you get there, and a large fairly close caribou herd to 1) keep your charter cost down, and 2) give you the best chance at taking a caribou. The caribou there are in velvet and are beautiful trophies in August. (Check with ADF&G for August bag limit.) The AK Pen herd also is known for producing very large trophy caribou. Check the AK map and locate KS about 300 miles SW of Anchorage at the NE head of Bristol Bay. The bulk of the caribou herd that time of year will be moving north somewhere above Port Heiden which is due south of KS. I've seen the herd even well north of Pilot Point in August and then you're in business with LOTS of places to land a plane. The plane you'll charter likely will be a Cherokee or C185 (both on larger 850 x 6 tires) and they can handle the sand blows. Most of these sand blows are huge. (You won't need a Super Cub for access.) Part of your advance planning is to be sure that your charter will land you off "runway." There are several remote strips in that area that Pen Air might want to limit you to. That will drop your success chances of taking a caribou to almost zero. Understand you can't "hunt" caribou when the herd is in migration. You have to be in front of the herd and let the caribou come to you. You have to know where the herd is and dropped off in front it to have any reasonable chance of success. Also, walking in that country is very difficult. I doubt you'll venture even a half mile from camp.
Well, I hope this helps you. I guarantee you would have a hunt to remember! A couple other pieces of advice: Watch the weather forecast. Always be prepared to be weathered in on the AK Pen for at least an extra week. The weather there can be bad. Your tent needs to be very high quality and able to shed wind, so I recommend one of the North Face dome tents or one similar. NOT a cheapy. Winds on the AK Pen ALWAYS are 20-30 mph that time of year and if a storm hits from the SE your winds likely will be above 75 mph. Also, no down clothing. Synthetic only. Finally, take a big enough caliber rifle with you and know how to shoot if VERY well. You're in brown bear country...very big brown bear. Oh, and take your fishing rod. Depending on the river, the fishing could be worth the trip!
Best of luck! PM me anytime.