No, we should not use genderless terms when referring to the God of the Bible. The Bible tells us that every word is inspired. Everything that God has given us is there for a reason. If we adopt the politically correct ideology and impose it upon the Bible regarding the use of the masculine, feminine, or neuter pronouns, then what is to stop others from altering other words in Scripture that don't suit their politically correct, gender equality perspective as they relate to things such as male eldership (1 Timothy 2:12-13)?
Furthermore, there are theologically important doctrines attached to the masculinity of God as revealed in Scripture. We have God the Father and God the Son. In the male gender, we have what I like to call, the gender of responsibility. That is, it is the male gender through which authoritative representation occurs. Let me explain.
In the Garden of Eden when Adam and Eve sinned, Eve sinned first. However, sin entered the world through Adam (Romans 5:12), not Eve. Why? Because Adam was the Federal Head. That is, he was the one who represented his descendants. More of this teaching can be found in such verses as 1 Corinthians 15:22 and Hebrews 7:7-10. In the representative nature of God the Son, where he represented us on the cross, it was the male gender that was chosen to bear our sins, die, and accomplish the redemptive work by which we are justified. From this understanding, we can also derive the doctrine of the Eternal Sonship of Christ. It was not the eternal "child ship" of Christ. It was not the eternal "daughter ship" of Christ. It is the Eternal Sonship of Christ. This is because Jesus is the Son of God, not the daughter of God.
Within the doctrine of the Trinity (three distinct persons, the Father, the Son, and the Holy Spirit), we find the inter-Trinitarian communion that has eternally existed. Within the doctrine of the Trinity, we can see the eternal covenant by which mankind would be saved. Part of that eternal covenant is that the Son, the second person of the Trinity, would become flesh and bear our sins. This eternal decree of God regarding the incarnation of Jesus was an eternal decree that the masculine gender would be chosen to be the Messiah, God in flesh. By necessity, this means that the male gender is the chosen gender of authority.
Male representative authority does not mean that the male is superior in nature to the female any more than it would mean the female is superior to the male because the female can bear children. With people, authority deals with position, not nature. In the Trinity, the Father sent the Son and the Son was made flesh (John 1:1,14) under the law (Galatians 4:4), and thus Jesus would say that the Father was greater than he (John 14:28). This is because Jesus was in a lesser position that God the Father, even though Jesus is called God the Son (John 1:1, 14; Colossians 2:9; Hebrews 1:8). So as the male gender he was under the law and in submission. But, because he's God in flesh, he is superior to us by nature and has the authority to represent us.
So, gender specific terms in relation to God are connected to theological doctrines. If those gender-specific terms are altered, then those related doctrines can be affected as well. This is why we should not compromise the word of God and subject it to gender-specific political correct ideology. The word of God is not to be altered by the whims of secular people. It is the other way around. The secular world, whether it admits it or not, is subject to the word of God. Therefore, the word of God stands secure and superior to man, including his politically correct views.
Don't alter the word of God to suit your preferences.