Is it permissible for a mother to grant her adopted son her house, while compensating her brother (her sole heir) with money, and would she bear any sin for doing so?

To begin with, the statement "I have no one but Allah and my mother" is incorrect because it implies a form of polytheism (shirk). The correct phrasing is: "I have no one but Allah, then my mother."

Regarding the gift (hiba), if the woman is not your biological mother and she gifted you from her property while she was in good health, and you took possession of the gift, it becomes your property, and her heirs have no share in it after her death. However, if she gifted it to you while she was suffering from a life-threatening illness (marad al-mawt al-makhouf), then the gift takes the ruling of a bequest (wasiyya) and is only enforceable within the limits of one-third (of the estate) or with the permission of the heirs. It is not valid for an heir unless the other heirs approve it.

And whatever she gifted to you and you took possession of while she was in good health, there is no prohibition on it, even if she intended to deprive her heirs. Although some scholars hold the view that she incurs sin for intending to deprive her heirs.