Giving money to those whose hearts are to be opened is not a bribe, because a bribe is money which a person gives to someone to help him to avoid a duty or to get something to which he is not entitled. But giving money in order to soften a person’s heart and open it towards Islam is helping them to discover the truth and encouraging them to follow it by entering Islam. It is a kind of jihad using money. 

Allaah has allocated a share of zakaah for those whose hearts are to be softened, as He says (interpretation of the meaning): 

“As-Sadaqaat (here it means Zakaah) are only for the Fuqaraa’ (poor), and Al-Masaakeen (the poor) and those employed to collect (the funds); and to attract the hearts of those who have been inclined (towards Islam)…”

[al-Tawbah 9:60]

This share should be taken to the ruler and given to those who he sees are close to accepting Islam. The Prophet (peace and blessings of Allaah be upon him) gave those whose hearts were to be softened wealth from the spoils of the battle of Hunayn, which led to entire tribes becoming Muslim. This is an ongoing issue, and one of the means of da’wah that should be revived, because people are naturally inclined to like those who are kind to them: 

“Be kind to people and you will capture their hearts

How often has kindness made people treat you like a master.” 

al-Qurtubi, al-Jaami’ li Ahkaam al-Qur’aan, 8/181; Ibn al-Atheer, al-Nihaayah fi Ghareeb al-Hadeeth, p. 359; al-Booti, Fiqh al-Seerah, p. 430 

The correct attitude for the Muslim, if he sees a clear and unambiguous text concerning some matter – whether it coincides with current custom, or with western culture, or not – is that he should follow it without any kind of embarrassment or fear, and whether he knows the wisdom behind it or not. 

May Allaah send blessings upon His noble Prophet.