This is a risky trick but luckily with more or less a fail-safe method. The start is usual: start Evidence Dash and reach on_mission == 0. After it the most risky trick has to be executed. The thing is that if you just enter the marker, one copy of Gangcar Round-Up will instapass (the one that was originally Evidence Dash), the other will still be running and since instapassing will set on_mission to 0 again, Kingdom Come will start (the player stays in the marker). And this would crash the game.

So how to avoid this? You have to activate mission while still moving and exit the marker by the end of the movement. However since the game activates this particular mission only when player is not moving (not using keyboard as far as I understand), there is only one way to execute this trick: by jumping off the phone. The most consistent way in terms of fail-safety in shown on video. Mission will start, one copy will instapass and the other will still be running. If may require a few tries but it shouldn't crash the game.

To cancel the other copy of mission you need to start vigilante. The next problem that now both Kingdom Come and Gangcar Round-Up are available. If you immediately enter the marker, first mission to start would be Kingdom Come with Gangcar Round-Up being second. As we know, this leads to the second of the missions to be active. However after reloading the game the order of mission-starting scripts will reverse. So Kingdom Come will be activated. It can be passed normally, however you have to start vigilante after its completion once again because one more copy of Kingdom Come will still be running.

This instapass works for 100% only since it doesn't affect mission counter. However the percentage is increased.

Also thanks to Powdinet for original discovery of instapass and video.

I might've found a way to skip the 'save & reload' part by 'exiting the marker while taking the mission' once more after starting Vigilante. It also doesn't start a second instance of Kingdom Come (so no need to start Vigilante once more before taking another mission as shown at the end of the video below).

Problem: Gangcar Round-Up may start instead of Kingdom Come (after Vigilante) and I have no clue how to control what mission starts.

EDIT: You don't even need Vigilante after instapass in case you get Kingdom Come, because you get a "Mission failed!" during the mission start cutscene and then you will be on mission = 0 for Kingdom Come; after finishing the mission there should be no remaining instance running.

EDIT #2: new video including a "solution" to the above problem, hold Vigilante submission, then if you get Gangcar Round-Up activate Vigilante after mission cutscene to abort the mission and try again, if you see Kingdom Come immediately let go of caps lock before mission cutscene starts, then proceed to do Kingdom Come normally.

Alternatively (not in the video) it could be fine to just deathwarp if you get Gangcar Round-Up and try again. The reasons why it isn't as bad as it sounds: the phone is close to the hospital, you don't need Vigilante (and a cop car) at all and you get onmission = 0 for Kingdom Come (which can lead to an easy ED dupe for example).