If you have congestion pricing, and the money is paid in cash, people who value the money more than the convenience will use the transit system earlier or later when it's less crowded, and pocket the difference.
Or is the lump sum grant based on something like the average price per ticket over both rush and non-rush hours? Then you can solve the crowding problem (by averaging appropriately to adjust the apparent cost of rush-hour transit), but you haven't solved the equality problem, because poor people won't be subsidized enough to take the transit during rush hour.