Is it permissible to sell the current equipment and divide the proceeds between the partner and the heirs of his deceased brother, or for the partner to buy some of the equipment for himself and give its value to the heirs when the partnership is dissolved?

One must be cautious about illicit gains in trade, such as selling internet services that contain Sharia violations, or selling media tools to those who use them for forbidden acts. As for the partnership, since the capital is from both partners and the work is from one of them, it is a partnership of inan (shared capital, one partner works), and the profit is divided according to the agreement. The share of the deceased partner is distributed to his heirs according to Sharia. You may take a fee for your management of the project. The permission of the heirs is required to trade with their deceased's money; otherwise, the partnership must be dissolved, and their shares given to them. Everything in the company is equally divided between you and the heirs, and if you wish to buy their share, the price must be estimated justly.