Is it permissible to sponsor an orphaned child (one year old) in the home when there is a daughter (12 years old), and is there a concern that the child might suffer from a psychological crisis when he grows up due to not being the biological son?

It is permissible for you to sponsor the orphan and raise him in your home, provided his guardian and mother give permission, and you will receive a great reward for that. To avoid awkwardness when he reaches puberty, you can breastfeed him now with a Sharia-compliant suckling so that he becomes your son through

fosterage and a brother to your daughter through fosterage.