After the bees have adjusted their GPS, and it is time to do the cutout (during good weather). Take the old mail box several meters away and replace it with a nuc box with at least one frame of brood whilst doing the cutout. The aggressive bees will give up on you and go back to the old site and populate the nuc. Once most of the field bees have gone back to the old site, it will be much easier to do the cutout with mainly nurse bees left.
Yes I would do a split if the population if big enough, depending on what time of the season you are in.
The nuc you create can be one split and the cutout the second.
Don't try to save every bit of the hive into empty frames, only the good worker comb. Some of that can go into the nuc.
Alternatively you could forget about the nuc box & let the bees fly around at the old site until you place the cutout box in place.