WARNING! Works only on a PC, it can crash the game and corrupt the save file on other platforms.
EDIT: There is a method now to obtains these cars without cheats. Check this video:
Thanks for Bender. and Patrick1994 from gtaforums.com for providing a method to obtain Picador. Video by Patrick:
I used mission duplication glitch and the Blood Bowl trick, then I played around for a few hours until I discovered a method to obtain this Fully Immune cars. You could probably do this without a cheat by timing your death by flames appropriately.
My controls are:
TAB - pick up the phone call, view stats
F - action/cancel button (jacking a vehicle, getting in the vehicle)
Space - accept/sprint button
Here is what you have to do:
1) Get a camera and a gun that you can walk with when you are aiming (pistol, Tec-9, Micro-SMG). You can also fail this Photo Opportunity mission once to get a Trip Skip.
2) You have to be after Deconstruction mission but before answering the phone call from Jethro
3) Get to the Blood Bowl Arena, aim with your weapon, remember the time and shoot. The phone call will be postponed by 15 seconds.
4) Stop aiming your gun, start aiming your gun again and walk in the marker.
5) When 15 seconds since your shot passed, stop aiming your gun, wait until the screen starts fading to black and hold down the escape button.
6) When you hear a ringing of your phone, release the escape button. If you are in the menu, press the button once to exit the menu.
7) Try to move your car during the cutscene. If it moves, then all is good. You can press the F button to skip a cutscene.
8) Before the timer runs out, press and hold TAB. You will be holding it for a long time, don't let go!
9) Still holding TAB, select your camera and run to the Photo Opportunity mission marker near the garage in Doherty. You can jump on cars to give you a lift. (Note: when I tried to get there by car, I couldn't enter the mission marker when I got out of the car)
10) After entering the marker, you can skip Cesar's talk by pressing F
11) Still holding TAB, enter a car. If everything went well, you should see a Save Game Icon and a mission marker, even though you are technically on a mission
12) Drive to Cesar or use Trip Skip. I don't know if you have to hold TAB during that, but better safe than sorry. I used Trip Skip, so it wasn't a big deal.
13) Drive with Cesar to Angel Pine or use Trip Skip again
14) Still holding TAB, enter the mission marker and get out of car. Make sure that you are holding your camera.
15) Go upstairs and enter the mission marker. You will be forced to pick up the phone. You can finally let go of TAB now. Press F to skip the phone call. If you have your camera selected, then press RMB. You should get locked in this "camera view mode".
16) Proceed through the mission like normal until Toreno arrives (if you want to have all the cars to choose, otherwise you can proceed to the next point after making a photo of Ryder or T-Bone Mendez if you just want AP ZR-350 or Washington respectively)
17) This part requires some speed. Immediately after taking a photo, type "SZCMAWO" to commit suicide. You should get wasted without failing the mission.
18) Sprint to the place of Loco Syndicate's gathering. You can press W + Space even slightly before you fully respawn to get a little more time.
19) Select a weapon and kill the syndicate's last member (Jizzy B in my case). This will freeze the mission.
20) Select your AP car and drive to Doherty's garage. I chose ZR-350, because I like it the most. There will be no vehicles or pedestrians.
21) Park your car inside the garage and get to the Driving School (by car or on foot, it shouldn't matter), enter it
22) Press and hold the F button for a second or two, then enter the marker
23) Release the F button while you appear back in the school, then press Space. I died this time for some reason, but usually I just appear with my parachute selected. I pressed Space when I respawned after my death. This should clear the frozen mission.
24) Get to your Doherty garage and get in your car. You can see that the engine is still running, so this car isn't really saved in the garage.
25) Drive to a better garage if you have one, otherwise just get out of car and leave it in Doherty (this garage might be a bit risky to use).
26) Save the game and enjoy your AP car! You can also check if the save doesn't become corrupted, it shouldn't happen though.
Note: It should be possible to get that cars without cheats, just with some precise death timing. I'm feeling tired of this mission already, so I probably won't test it further. I leave it to others.