Is it permissible not to have a marriage banquet if holding it would lead to religious violations such as intermixing between men and women, illegal exposure of women's adornment, singing, and so on?
Can I make up for this by slaughtering a sheep for example and distributing the meat among my relatives and the poor people?
May Allaah reward you.

All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

If one is sure that hosting a marriage banquet will lead to people committing prohibitions , then he must not hold it for two reasons:

1-      Holding a marriage banquet is not obligatory according to the preponderant opinion, rather, it is an act of Sunnah (Prophetic tradition) or a recommended act.

2-      Holding a marriage banquet and inviting people to it are good pursuits, while the expected religious violations are evil and it is known that avoiding evil takes precedence over achieving interests.

Nevertheless, a Muslim should be strong in his adherence to the truth. Hence, if you are the one who will organize the banquet, you can do so without allowing those whose presence will lead to committing prohibitions.

In this way, you would probably encourage others to imitate you. In such a case, you will be rewarded for reviving an act of Sunnah and you will also receive a reward equal to those who will imitate you, either directly or indirectly, until the Day of Judgment.

However, if you fail, then it would be better for you to slaughter a sheep and invite your virtuous friends to it with the intention of having a marriage banquet. After that, you can distribute the remaining meat to your friends, relatives and the poor and announce that this is the food of the marriage banquet.

Allaah Knows best.