To equip the original prop, equip the "Sharlayan Coffee Cup" prop listed in Anamnesis. For the Last Stand cup, change the 1 in the third box to a 3. With this mod installed, this cup will be dyeable, with the dye also affecting the colour of the Last Stand logo.
The minion prop is not directly dyeable (as minions cannot be dyed), but the colourset is easily edited to match whatever dye your heart desires.