I do use a laptop occasionally but I've not had any of the issues you mentioned in the other thread. Although my laptop also costs 3x as much as you want to spend, but I don't think that's the issue.
I could imagine that the issues arose due to laptops being often quite aggressive in terms of power-saving, especially when running on battery. They often start to underclock everything and most laptops come with two graphics chips nowadays and you have to specifically tell it to use the powerful chip instead of the onboard one which usually is just enough to run the most basic graphics stuff.
I can't really give any specific recommendations. There's more to it like size/weight preferences, how much you plan on traveling etc. I personally have an MSI, but realistically Construct should work well on pretty much any random laptop that isn't ancient.