Is the first party entitled to claim profits for the period of the dispute after the second party continued to invest the money and refused to pay the profits on the grounds of the dispute?

Your agreement with the investor to guarantee the principal and the possibility of its retrieval within a month makes the transaction impermissible. This is because in a valid Mudarabah (definition) (Islamic partnership), it is not permissible to stipulate a guarantee of the principal; otherwise, it would transform into an interest-based (usurious) loan. In this case, the principal and all profit belong to you, and the loss is yours, while the Mudarib (investor/partner) is entitled to a fair wage ( اجر المثل). If the capital is readily available or goods that can be sold without loss, and the investor refuses to hand it over, he becomes an usurper and bears the loss. However, if the goods can only be sold at a loss, the invalid Mudarabah remains in effect, the principal and profit are yours, the loss is yours, and the Mudarib is entitled to a fair wage.