What is the ruling on adopting a child born out of wedlock and raising him Islamically, given the complexities of annulling the adoption and the potential marital problems this might cause?

Islam has forbidden adoption because it infringes upon rights and mixes lineages. Adoption must be abolished if possible without causing harm. However, if its abolition would lead to harm, then there is no objection to maintaining it, provided a document is drawn up to prevent any religiously forbidden matters. You should inquire about the Islamic ruling before taking any actions.