Can a 5 Point Calvinist Be A Baptist Fundamentalist?

Depends on your definition of a fundamental

 

The FBFI Constitution says nothing about soteriology.

 

To use a bit of historical shorthand, for the most part, the Fundamental Baptists are Particulars who follow the Andrew Fuller school of thought. So, we don't make a big thing of "The Doctrines of Grace". That's seen as a Southern Baptist kind of thing.

 

Would you agree these 5 doctrines are the fundementals:
Christianity[edit]
Main article: Christian fundamentalism
Christian fundamentalism has been defined by George Marsden as the demand for a strict adherence to certain theological doctrines, in reaction against Modernist theology.[12] The term was originally coined by its supporters to describe what they claimed were five specific classic theological beliefs of Christianity, and that developed into a Christian fundamentalist movement within the Protestant community of the United States in the early part of the 20th century.[13] Fundamentalism as a movement arose in the United States, starting among conservative Presbyterian theologians at Princeton Theological Seminary in the late 19th century. It soon spread to conservatives among the Baptists and other denominations around 1910 to 1920. The movement's purpose was to reaffirm key theological tenets and defend them against the challenges of liberal theology and higher criticism.[14]

The concept of "fundamentalism" has roots in the Niagara Bible Conferences that were held annually between 1878 and 1897. During those conferences, the tenets considered fundamental to Christian belief were identified.

"Fundamentalism" was prefigured by The Fundamentals: A Testimony To The Truth, a collection of twelve pamphlets published between 1910 and 1915, by brothers Milton and Lyman Stewart. It is widely considered to be the foundation of modern Christian fundamentalism.

In 1910, the General Assembly of the Presbyterian Church identified what became known as the five fundamentals:[15]

• Biblical inspiration and the infallibility of scripture as a result of this
• Virgin birth of Jesus
• Belief that Christ's death was the atonement for sin
• Bodily resurrection of Jesus
• Historical reality of the miracles of Jesus

In 1920, the word "fundamentalist" was first used in print by Curtis Lee Laws, editor of "The Watchman Examiner," a Baptist newspaper.[16] Laws proposed that those Christians who were fighting for the fundamentals of the faith should be called "fundamentalists."[17]

Theological conservatives who rallied around the five fundamentals came to be known as "fundamentalists". They rejected the existence of commonalities with theologically related religious traditions, such as the grouping of Christianity, Islam, and Judaism into one Abrahamic family of religions.[3] By contrast, while Evangelical groups (such as the Billy Graham Evangelistic Association) typically agree with the "fundamentals" as they are expressed in The Fundamentals, they are often willing to participate in events with religious groups which do not hold to the essential doctrines.[18]

 

I can only speak to the roots of the FBFI. It was born out of the Modernist controversy in the Northern Baptist Convention in the 1920s. The then FBF was a meeting of like-minded individuals before the annual meeting of the Convention. It continued inside the NBC until the late 40s when its membership split off into the Conservative Baptist Association. The FBF however still met under the CBA umbrella. Until that is the mid-1960s with the rise of Neo-Evangelicalism and the synchronistic Billy Graham Crusades.
So, the answer to the question of the relationship of the FBFI to the Doctrines of Grace is found in its roots in the Andrew Fuller branch of the Particular Baptists which formed the Triennial Baptist Convention for Missions in the early 1800s.