Do conservative Presbyterians accept 4-point Calvinists?

Conservative Presbyterian churches typically require their elders to subscribe to the Westminster Standards, which include the Westminster Confession of Faith, the Westminster Shorter Catechism, and the Westminster Larger Catechism. The lay members of the congregation are usually only required to submit to the teaching of the elders, which in practice means that a wider variety of beliefs are tolerated than what the Westminster Standards permit.

In addition, many Presbyterian denominations permit the taking of exceptions to the wording or entire articles of the Westminster Standards. The extent and permissibility of these allowed exceptions varies from denomination to denomination.

The Westminster Standards do not align with the Amyraldian position, so a potential office-bearer would not be able to honestly subscribe to the Westminster Standards and remain Amyraldian. However, there are some positions of Calvinistic hypothetical atonement which some argue are plausibly acceptable within the bounds of the Westminster Standards, such as those promoted by James Ussher and Richard Baxter.

The relevant sections in the Westminster Confession of Faith are Chapter 3, Article 6, and Chapter 7, Article 3. The relevant Westminster Larger Catechism question is Question 67. I quote them below:

Chapter 3, Article 6. As God has appointed the elect unto glory, so has He, by the eternal and most free purpose of His will, foreordained all the means thereunto.Wherefore, they who are elected, being fallen in Adam, are redeemed by Christ, are effectually called unto faith in Christ by His Spirit working in due season, are justified, adopted, sanctified, and kept by His power, through faith, unto salvation. Neither are any other redeemed by Christ, effectually called, justified, adopted, sanctified, and saved, but the elect only.

Chapter 6, Article 3. Man, by his fall, having made himself incapable of life by that covenant, the Lord was pleased to make a second, commonly called the covenant of grace; wherein He freely offers unto sinners life and salvation by Jesus Christ; requiring of them faith in Him, that they may be saved, and promising to give unto all those that are ordained unto eternal life His Holy Spirit, to make them willing, and able to believe.

Q. 67 What is effectual calling?

A. Effectual calling is the work of God’s almighty power and grace, whereby (out of his free and special love to his elect, and from nothing in them moving him thereunto) he doth, in his accepted time, invite and draw them to Jesus Christ, by his Word and Spirit; savingly enlightening their minds, renewing and powerfully determining their wills, so as they (although in themselves dead in sin) are hereby made willing and able freely to answer his call, and to accept and embrace the grace offered and conveyed therein.

Double predestination is more clearly supported in the Westminster Standards.

Chapter 3, and in particular Article 3 and Article 7 are clearly in support of double predestination:

Chapter 3 Article 3. By the decree of God, for the manifestation of His glory, some men and angels are predestinated unto everlasting life; and others foreordained to everlasting death.

Chapter 3 Article 7. The rest of mankind God was pleased, according to the unsearchable counsel of His own will, whereby He extends or withholds mercy, as He pleases, for the glory of His sovereign power over His creatures, to pass by; and to ordain them to dishonor and wrath for their sin, to the praise of His glorious justice.

In answer to the involvement permitted, a conservative Presbyterian congregation or denomination would most likely not allow the ordination of someone who denies double predestination, but would potentially allow an exception taken on Amyraldianism, or alternatively the more permissible positions of Ussher and Baxter. Ordinary members would be otherwise comfortable, depending on their understanding of their membership oath to submit to the teaching of the elders of the church. Each congregation/presbytery/denomination will vary somewhat, though. Your mileage may vary.

There are two articles by Lee Gatiss, titled Shades of Opinion with a Generic Calvinism: The Particular Redemption Debate at the Westminster Assembly, and A Deceptive Clarity? Particular Redemption in the Westminster Standards which give a brief summary of what occurred during the debate of the Westminster Assembly on the issue of hypothetical universalism. The minutes of the debate are also available.