Do the partner's actions of evading the written documentation of the partnership, not paying enough money for the work, evading giving the other partner his dues, and withdrawing money without agreement, remove the Messenger (ﷺ) from being the third of the two partners, and are these actions legitimate?

The text of the hadith mentioned in the question: "Allah says: 'I am the third of the two partners as long as one does not betray the other. If he betrays him, I depart from between them,'" is weak.

The matter of partnership should be documented, as Allah Almighty says: {And take witnesses when you conclude a contract; and let no scribe be harmed or any witness} [Al-Baqarah:282].

Evasion and prevarication by a partner is a trick to seize a share, and the issue must be promptly brought before the Sharia courts.

Assuming the hadith is authentic, the third partner is Allah Almighty, and the meaning of Allah's partnership is to bestow blessing and profit upon the money. He is with them through preservation, blessing, sustenance, and goodness in their dealings.