You will get the login bonus from whatever it was when you start the race. If you go multiple days without going to the main menu, you will not accumulate more bonus (if you started at 150% bonus and drive for 4 days, you will still get 150%). HOWEVER, after the race, when you finally make it back to the main menu, the bonus will reset to 110%.
That being said, you might be able to keep the bonus with a little bit of save file swapping (I haven't personally tried this). If you have a USB flash drive, make a copy of your save file before you start the race onto it and move that Save onto your computer. Each day, copy the race save file to your computer (DON'T OVERWRITE) and copy the pre-race save file back onto your Playstation. Start it up and get to the main menu to get the log in bonus, then switch the save files back and continue your race. I believe the log in bonus is server-based, so as long as you can get to the Main Menu, it should keep going up.