The specs required for these games is low, fortunately. Cheaper gaming laptops typically go for twice your max budget. Looking for a used laptop is the way I would go with the budget. I have the specs for fortnite below, but keep in mind, these are the minimum expectations not what you should get. I would search for something with 1.5-2x these specs if you expect your kid to have the laptop for more the 3-4 years and the demands of future games.
https://www.chillblast.com/blog/fortnite-pc-specs-requirements-2024
CPU - Core i3-3225 3.3 GHz. GPU - Intel HD 4000 or AMD Radeon Vega 8. Memory - 8 GB RAM. Storage - 30GB available space