I used to lube the cases, but after I got a set of carbide .40/10 dies, I started resize the cases with that resizing die first. Then no lube is needed for resizing the neck with the steel .357 Sig die.
Typically, I'll resize/deprime with the .40 die, then switch to the 4 hole turret which holds the Lee 3 die .357 Sig set, and a FCD. I'm not familiar with the Lee 3 hole turret, but if it has removable heads, you may want to buy a dedicated carbide resizing die in .40/10. Then you can resize the body and neck as a batch, and switch to loading cases with a different head, which could hold the flare die, seating die, and a FCD if you feel you need one.
Speaking of the FCD, it's not entirely necessary. If you only flare the case a tiny amount (or not at all), it's possible to hold the bullet on the case mouth and carefully ride the ram up to the die. There may well be enough neck tension that the FCD isn't necessary. I use mine anyway, but very sparingly, because too much will crimp the bullet and ruin the neck tension causing setback issues.