This method was discovered by bug777. You can obtain AP Packer and DP/EP/FP BF-400 with this method as well, but these vehicles didn't concern me. I only cared about my beloved AP Sultan.
This method is much easier than the Info Icon Method. Here's what you have to do:

1) Grab some satchel charges from Montgomery. I recommend getting a body armor and doing some stamina training as well.
2) Start the mission and try to keep Big Smoke from getting shot too many times while not losing too much health yourself
3) When you get to the rooftop, kill some Russians, but leave 2-3 alive. Plant a satchel charge behind each one of them (as far away from Smoke as possible).
4) Jump off the rooftop, sprint to the vending machine near the gas station, push the button on the detonator and buy a drink immediately after.
5) The next part of the mission will look normally, but you will be able to jump off the bike at any moment. Try to get the bike's health bar to the low level before the cutscene with a truck falling off the bridge.
6) Soon after the cutscene, jump off the bike and follow the truck. The bike should get destroyed and the mission will fail.
7) Kill the Russians shooting at you and get in one of the Sultans that were on the Packer. I recommend the red one since it is hard to mistake with other (not AP) Sultans.
8) Test the Sultan to see if it's AP, drive to the garage and save. Enjoy this awesome car!

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