The Winter Watchtower and the Ice Cream Watchtower did not work for me to receive the special drops.
I can confirm that the Toadstool Adventure works for the sword!
It also took me 3 Watchtowers to get the drop, but I received it from the Gingerbread one!
For anyone who is wondering - to collect the arrows you don’t have to fight any monsters. You just collect from farmhouses. In 13 collections I got 3 arrows.