I've so far found this can be done in three places; at the start of a date, which is the beginning of the "date glitch", and during the missions "Smackdown" and "Pest Control".

-I don't think it's possible to shoot your weapon while the menu is up -After passing a mission, the end of mission call may not come in until you reach a certain spot on the map if the police menu is still up (Smackdown) -Smackdown's end of mission call always came in when I hit the closest intersection to the NW of Packie's house and at this point I was able to use the phone (make and receive calls) -You're able to receive and make mission triggering calls while the police menu is up after setting up the glitch with a mission or any other unknown way -If you're not already on the vigilante menu when one of these calls is triggered, you won't be able to access the vigilante missions -Friends don't seem to want to get into your police vehicle while the menu is on screen -I haven't found a mission marker that will trigger while the menu is on screen -The police vehicle is indestructible while the menu is on the screen -Niko is not invincible and will take damage as usual, though it sometimes seems like you can take more than you should be able to

Triggering the glitch on Smackdown: Search for Bucky out front of the Burger Shot > move to the marker and cancel the auto-call with E > when the marker triggers, press E again to bring up the menu > skip the cutscene with Enter

Triggering the glitch on Pest Control: Sit next to the curb, in the crosswalk > receive Pegorino's call > answer and end the call > quickly bring up the police computer > skip the cutscene

Indestructible vehicle, a use of this glitch (as well as its set-up during Pest Control):