What is the cheapest laptop that can run UE4?
Consider using remote desktop to run UE4 on a nice desktop computer, but use it from a laptop. Not quite as responsive as being there in person, but you have all the functionality as long as you have internet connection.
It hard to say, you should look on specs of laptop. At very least 8GB or 16GB RAM and AMD Ryzen APU as Intel+Nvidia combo is know to cause issues, the more cores the better.
But considering you want to buy hardware, i really recommend you to get desktop PC as lot laptops are under powered then desktop counterparts, and you can get desktop cheaper then laptop of same power, you technically paying for mobility, or else you really want that mobility, but if laptop will sit in same place all the time there really no point.