Is it permissible for my father to use his share from the sale of the bakery, for which he paid one-third of the capital in cash while his uncle paid the remaining two-thirds with an interest-based loan, to perform the rituals of Hajj?

The impermissibility of a usurious loan is related to the borrower's liability, not to the amount of the loan. Therefore, there is no objection to the capital of the aforementioned project or its profits. Even if one were to say that the impermissibility is related to the amount of the loan, there would still be no objection to the father performing Hajj with his entitlements from that project, because he performed Hajj with his own money and profits.