I can't help but think this may be due to a misunderstanding of what it means to pray "in the Name of Jesus."Plenty of unregenerate people praying in the name of Jesus.
It does not mean we must end every prayer with "in Jesus Name."
The phrase "in the name of" is a way of stating authority. For instance, those of us who are old enough remember the old Keystone Cops serials at the movies on Saturday. The inept cops would chase a criminal while yelling "Stop in the name of the Law!" Nobody assumed the cop was named John Law. It was a statement of authority. "By the authority of the law I demand you stop" would be a good paraphrase.
When we pray "in Jesus' Name" we are praying with His authority. Matthew 28:18 tells us "Jesus came to them and spoke to them, saying, “All authority has been given to me in heaven and on earth.""
Jesus told us, in John 14:13 "And whatsoever ye shall ask in my name, that will I do, that the Father may be glorified in the Son."
This is a statement of authority. Jesus tells us that, if we are in Him, and He is in us, we can ask using His authority, for our prayers to be answered.
For those not in Christ there is no such authority. We have fellowship (in prayer) with the Father because of our position in Christ.
1 John 1:3 that which we have seen and heard we declare to you, that you also may have fellowship with us. Yes, and our fellowship is with the Father, and with his Son, Jesus Christ.
"In Christ" = pray with Jesus' authority to the Father with access due to our position in Him.
"Not in Christ" = no such authority to address the Father regardless of what words are tacked onto the end of the prayer.