Does the Presbyterian Church in America (PCA) have a policy regarding admitting members who do not accept its doctrines?

The Presbyterian Churches of America have in their church order a section on membership. This does not require the members to adopt all the doctrines of the church, but it does require a credible profession of faith in Christ.

CHAPTER 6

Church Members

6-1. The children of believers are, through the covenant and by right of birth, non-communing members of the church. Hence they are entitled to Baptism, and to the pastoral oversight, instruction and government of the church, with a view to their embracing Christ and thus possessing personally all benefits of the covenant.

6-2. Communing members are those who have made a profession of faith in Christ, have been baptized, and have been admitted by the Session to the Lord's Table. (See BCO 46-4 for associate members).

6-3. All baptized persons are entitled to the watchful care, instruction and government of the church, even though they are adults and have made no profession of their faith in Christ.

6-4. Those only who have made a profession of faith in Christ, have been baptized, and admitted by the Session to the Lord's Table, are entitled to all the rights and privileges of the church. (See BCO 57-4 and 58-4)

To clarify on "a profession of faith in Christ", the church order states that the following questions must be agreed upon by the potential member to constitute a profession of faith in Christ:

1. Do you acknowledge yourselves to be sinners in the sight of God, justly deserving His displeasure, and without hope save in His sovereign mercy?
2. Do you believe in the Lord Jesus Christ as the Son of God, and Savior of sinners, and do you receive and rest upon Him alone for salvation as He is offered in the Gospel?
3. Do you now resolve and promise, in humble reliance upon the grace of the Holy Spirit, that you will endeavor to live as becomes the followers of Christ?
4. Do you promise to support the Church in its worship and work to the best of your ability?
5. Do you submit yourselves to the government and discipline of the Church, and promise to study its purity and peace?

In addition, to become a communicant member (that is, to participate in the Lord's Supper), the church order does stipulate that:

58-2. The ignorant and scandalous are not to be admitted to the Lord's Supper.

This would suggest that if the potential member has no knowledge of the doctrines of the church, then they would be ignorant and thus not to be admitted to the Lord's Supper, also excluding them from church membership.

There is also the matter of what constitutes an offence worthy of church discipline. The church order states the following:

29-1. An offense, the proper object of judicial process, is anything in the doctrines or practice of a Church member professing faith in Christ which is contrary to the Word of God. The Confession of Faith and the Larger and Shorter Catechisms of the Westminster Assembly, together with the formularies of government, discipline, and worship are accepted by the Presbyterian Church in America as standard expositions of the teachings of Scripture in relation to both faith and practice. Nothing, therefore, ought to be considered by any court as an offense, or admitted as a matter of accusation, which cannot be proved to be such from Scripture.

This suggests to me that, if a person were to become a member of the church who did not hold to the doctrines maintained to be true by the confessions of the church, they would immediately be subject to church discipline as offending through false doctrine.

It is possible that this is not typically exercised in this way by many/most PCA church sessions. For your particular case I would ask the session of your church what they recommend.