I want to move some sims out of a household into another lot. I have enough money, but the game insists on splitting the money 'fairly', so e.g. if I have 8000$ and 6 sims, it only allocates a bit more than 1000 to the sim moving out and won't let me buy a lot that costs 7000$. Am I remembering wrongly that this used to be possible? Or might it be a side effect of some mod I have?

There's a lot of ways to manipulate these things as the player. Without changing any settings, what I would probably do is MC > Add Sim the one breaking away to their new home lot, then do cash transfers between the two households if things remain out of balance (but be prepared to quit without saving if you get unexpected results when it comes to things like the family inventories).

Another workaround is to pretend like you're moving all the sims, so they have their full budget available; choose the new lot; then move all the sims you want to stay in the current house back to the left side of the window. The game will split the leftover cash equitably (in its estimation) among the sims and will let the current house selection stand.