I can see why the OP feels conflicted concerning this issue.
Yes, theologically, you only need to be Baptised once. However, sadly, separate denominations of Christian Church do not appear to consistently recognise the authority of each other's church - which can cause problems if you want to worship within a variety of different church denominations or bring your child up to know a variety of Christian denominations.
Church divisions and the problems it causes has been going on right from the beginnings of the church. Yet the Bible shows how church unity between all the body of Christ is important:
Galatians 3:27-29 New International Version (NIV)
"27 for all of you who were baptized into Christ have clothed yourselves with Christ. 28 There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus. 29 If you belong to Christ, then you are Abraham’s seed, and heirs according to the promise."
1 Corinthians 1:9-11 New International Version (NIV)
"9 God is faithful, who has called you into fellowship with his Son, Jesus Christ our Lord.
A Church Divided Over Leaders
10 I appeal to you, brothers and sisters,[a] in the name of our Lord Jesus Christ, that all of you agree with one another in what you say and that there be no divisions among you, but that you be perfectly united in mind and thought. 11 My brothers and sisters, some from Chloe’s household have informed me that there are quarrels among you."
I don't have any specific advice, only that the divisions are not where you find God, as unity is what the church should be aiming for. Ultimately is is your child who will be acting according to their own beliefs and what they are comfortable with.