Is paying an official a sum of money to obtain approval to demolish an old building on my private land, after the Antiquities Department's refusal, considered bribery and a sin?

What you mentioned regarding the competent authorities preventing the issuance of a building permit because the existing structure is considered an artifact falls under the ruler's restriction of what is permissible (mubah). The ruler has the right to restrict what is permissible if there is a public interest in doing so.

If the prohibition is not for a clear public interest, then there is no sin upon someone who obtains a building permit by paying money to someone who assists them. However, if the prohibition is for a clear public interest, then one is not allowed to circumvent it by offering money or other means. The estimation of benefits and harms is referred to those with experience and expertise.