Is it permissible for a person to refuse a DNA test to prove the paternity of a child born from a previous illicit relationship, or to offer a bribe to alter the result if paternity is proven, out of fear of destroying his current family?

If the situation is as described, then the child is not attributed to the man; rather, he is a child of adultery and is attributed to his mother. The mentioned test neither proves nor disproves lineage, nor does it change the ruling that he is a child of adultery. The man must employ stratagems to prevent the child from being attributed to him. If what the imams have suggested is a means to achieve this, then there is no harm, for what is given to attain a right or repel injustice is not considered a bribe on the part of the giver. A believer should not do anything that humiliates himself.