Hi:
If there is a Presbyterian Church nearby that would be willing to baptize your baby, then you should see if they would do it:
The grace which is exhibited in or by the sacraments rightly used, is not conferred by any power in them, neither doth the efficacy of a sacrament depend upon the piety or intention of him that doth administer it: but upon the work of the Spirit, and the word of institution, which contains, together with a precept authorizing the use thereof, a promise of benefit to worthy receivers, WCF 27:3.
There are only two things which make the Sacrament valid: The work of the Spirit, and the word of institution. In Baptism the word of institution is found in the Scripture, "I baptize you in the name of the Father, Son, and Holy Spirit." As long as a church uses the words of institution the baptism is valid - despite its doctrine, piety, or intention. Because the efficacy of it is found in the Spirit of God working with the Word of God.
Baptism can be performed at any time, and not only during worship service. Philip baptized the Ethiopian at a time that was not formally a worship service. As long as there is a Church officer who has the power to baptize (like Philip) baptizm can be done with/without witnesses, in church, out of church, or anywhere, or anywhen, a person wishes.
Blessings,
-CH