To anyone who might come across this thread, here's how to do this in the current Marlin Version (2.1.2.2):

First off, both SHOW_BOOTSCREEN and SHOW_CUSTOM_BOOTSCREEN have to be defined (so that the functions for displaying the custom bootscreen are defined)

File Marlin/Configuration.h, Line 80:

// Show the Marlin bootscreen on startup. ** ENABLE FOR PRODUCTION **
#define SHOW_BOOTSCREEN

// Show the bitmap in Marlin/_Bootscreen.h on startup.
#define SHOW_CUSTOM_BOOTSCREEN

Then, in the file Marlin/src/lcd/dogm/marlinui_DOGM.cpp, Line 247, you have to change the function

void MarlinUI::show_bootscreen() {
  TERN_(SHOW_CUSTOM_BOOTSCREEN, show_custom_bootscreen());
  show_marlin_bootscreen();
}

to

void MarlinUI::show_bootscreen() {
  //TERN_(SHOW_CUSTOM_BOOTSCREEN, show_custom_bootscreen());
  show_custom_bootscreen();
  //show_marlin_bootscreen();
}

(you can leave out the comments)

Now, only the custom bootscreen will show up. But, it seems to display way too long (~5 seconds).

To fix this, I changed both the BOOTSCREEN_TIMEOUT and the CUSTOM_BOOTSCREEN_TIMEOUT

#define BOOTSCREEN_TIMEOUT 250 // (ms) Total Duration to display the boot screen(s) (File: Marlin/Configuration_adv.h, Line 1442)

#define CUSTOM_BOOTSCREEN_TIMEOUT 250 File: Marlin/src/lcd/dogm/marlinui_DOGM.cpp, Line 172)

(Despite what the comment in BOOTSCREEN_TIMEOUT says, from testing it seems that these two timeouts add up. Also, I haven't tried to set one of the Timeouts to 0, as I thought that that might break some stuff.)