Is it obligatory to pay the deferred dowry that the father agreed to out of embarrassment, and is it affirmed in Islamic Sharia, and what if it is demanded from us, are we obligated to pay it?

There is no harm in the spouses agreeing upon a prompt dower and a deferred dower, according to their mutual consent. If the wife is divorced before consummation, she is entitled to half of the agreed-upon dower (both prompt and deferred), unless one of the parties relinquishes their right, based on the Almighty's saying: "And if you divorce them before you have touched them and you have already specified for them an obligation [dower], then for them is half of what you specified, unless they [the women] give up [their right], or he in whose hand is the marriage contract gives up [his right]. And that you give up [your right] is closer to righteousness. And do not forget generosity among yourselves. Indeed, Allah is Seeing of what you do." However, if she is divorced after consummation, she is entitled to the full dower, both prompt and deferred.