Is it permissible for the legal guardian to recover the sum of 500 dinars that he paid from his personal money to settle the deferred dowry of his deceased brother's wife, out of concern to prevent the sale of the residential house left by his brother?

If you were not a donor of the amount you paid to settle your brother's debt, then you have the right to reclaim this amount from his estate. Whoever fulfills an obligation on behalf of another without intending it as a donation has the right to seek reimbursement from them, even if the fulfillment was without their permission, such as paying off someone else's debt.