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.
Apparently, there is no consequent harm in giving your wife’s last name as a middle name to your child in terms of proving the paternity and the pertinent religious rulings, such as inheritance, unmarriagability and the like. Hence, this does not fall under the heading of the child bearing the name of other than his biological father, especially since the child bearing the name of his maternal grandfather is lawful in some respects.
Some Muslim scholars bore the names of their maternal grandfathers, given the fame of those grandfathers. The well-versed scholar At-Taahir ibn ‘Aashoor wrote in his book At-Tahreer wa At-Tanweer: “As for the verse that reads (what means) {Attribute them to their (biological) fathers …}, the pronoun ‘them’ refers to the adopted sons. The command does not include attributing the child to his grandfathers because they are considered their fathers. The Prophet, sallallaahu ‘alayhi wa sallam, said about Al-Hasan (ibn 'Ali) : ‘This son of mine is a master.’”
Shaykh Ibn Baaz said, “The daughter’s sons are called the sons of their maternal grandfather. It has been authentically reported that the Prophet, sallallaahu ‘alayhi wa sallam, said about Al-Hasan ibn ‘Ali: ‘This son of mine is a master.’ Al-Hasan was his daughter’s son. Allaah, The Exalted, stated that ‘Eesa (Jesus), the son of Maryam (Mary), was from the offspring of Prophets Nooh (Noah) and Ibraaheem (Abraham); Allaah, The Exalted, says (what means): {And We gave to Ibraaheem, Ishaaq (Isaac) and Ya‘qoob (Jacob) - all [of them] We guided. And Nooh (Noah), We guided before; and among his descendants, Daawood (David) and Sulaymaan (Solomon) and Ayyoob (Job) and Yusuf (Joseph) and Moosa (Moses) and Haaroon (Aaron). Thus do We reward the doers of good./And Zakariyya (Zechariah) and Yahya (John) and ‘Eesa (Jesus) and Elyaas (Elias) - and all were of the righteous.} [Quran 6:84-85]
It is well-known that ‘Eesa was born to a mother only and had no father. He was the descendent of Aadam (Adam), Nooh and Ibraaheem, may Allaah exalt their mention. As to the verse that says (what means): {… And [also prohibited are] the wives of your sons who are from your [own] loins…} [Quran 4:23] This is meant to exclude foster sons who were adopted in the pre-Islamic era. Allaah, The Exalted, made it forbidden; He says (what means): {Call them by [the names of] their fathers. It is more just in the sight of Allaah.} [Quran 33:5]”
This is the basic principle in this regard. However, in your case, the problem is that your wife’s last name is a Christian name. It is impermissible for the Muslim father to give up his due right to have his own son attributed to him and bear his “Muslim” name, which indicates his Islamic identity and belonging, and give him a Christian name. In fact, this is degrading. His name would not then be distinguished from Christian ones. This may constitute an imitation of the non-Muslims and the adopting of their habits and yielding to them.
Shaykh Bakr Abu Zayd wrote: "The Islamic Sharee‘ah underlined that it is impermissible to give the Muslim newborn a name that falls under the following categories … a non-Muslim name that is peculiar to non-Muslims … The Muslim whose heart is content with his religion avoids and refrains from taking such names … This is considered one of the pits of sin that incurs being forsaken by Allaah. Examples of those names are Peter, George, Diana, Rose, Suzan, and the like of such names. Taking the names of non-Muslims is considered an act of imitation. If it is done out of the person’s personal inclinations or indifference, then it is considered a grave and serious sin. If it is done out of belief that such names are superior and better than the Islamic names, then such a person is in great danger that imperils the fundamentals of his faith. In both cases, the person should not hesitate to repent, and changing such names is one of the conditions of the repentance.” [Tasmiyat Al-Mawlood]
Ibn Al-Qayyim wrote, “Names can be classified into three categories; some names are peculiar to Muslims, others are peculiar to non-Muslims, and a third category of names are shared by both parties. Examples of the names peculiar to non-Muslims are George, Peter, and John. It is impermissible for Muslims to take such names because it involves imitating the non-Muslims in their peculiar customs.” [Ahkaam Ahl Ath-Thimmah]
In addition, the name of the child’s grandfather (Alvarez) has nothing to do with the name Ibraaheem. Giving the child the mother’s last name as a middle name may deceive people into assuming that he (Alvarez) is his (Ibraaheem's) son; thus he will be attributed to other than his father and this is prohibited.
Finally, we advise the dear questioner to strive to invite his wife to Islam and encourage her to embrace Islam instead of fulfilling her wish of giving her Muslim son this non-Muslim name.
Allaah Knows best.