Don’t spread the budget for the whole month. You can still bring in good results but let them know that budget will not last a whole month.
Remove weekends / and maybe only run ads for a few hours a day even, but run them so you can get a good amount of traffic. Whatever it takes to show them results.
Then you can show them performance and have data on your side to further push the budget issue to them and help them realize more budget = more profit.