Is there a name for a category of Christians who lack a doctrinal position in the debate about God's nature?

Christians have been around for 2000 years. We would think surely there is already a label out there that answers your question?

Short answer: Deist is the closest, with Thomas Jefferson as a famous practitioner. No doubt to himself and others, Thomas Jefferson refer to himself as a "Christian", but because this conception of God is detached from day to day Christian practice of love, Deists are not truly Christians.

Answer

There is no such label, since someone who has that skeptical state of mind cannot practice a Christian life worthy of its name, since fundamentally the true Christian life is a costly yet life-saving response to a powerful yet loving being who offers a passage to heaven (salvation). Without being sure of God's nature how can a Christian trusts his life on this being called God? Therefore, skepticism about the nature of God cannot coexist with a flourishing faith life.

Two tracks: Academic vs. Life

One can definitely engage in the academic discipline of theology and philosophy to discuss the merits of various conceptions of God's nature: what are the Scriptural data, the early church fathers writings, the experience of saints, various councils, philosophical terms appropriate to define it, how God's attributes are affected by a definition of God's nature, etc. But this exercise is on a completely different track.

Consider the parallel track of LIFE. When this theologian stands up from the theologian's "arm chair" and deals with his wife, kids, work colleagues, bills, politics, his health, culture, etc. he needs to presume with full confidence a certain understanding of God's nature so he can interface with the people around him as a Christian.

Christian life presumes certainty of the nature of God

Why is it that in this track one cannot be skeptical? Because underlying the love he has for his wife, kids, and neighbors, as a Christian he has to embody (or to image) God's nature in a human form just as Jesus incarnated God's nature as a perfect Adam: a loving, generous, self-sacrificing, uplifting, servant-leader person. That's what it means fundamentally to be a Christian.

God is love. Christianity is all about love. Mormons practice love, although their understanding of God is not standard Trinitarian, because they don't believe that the 3 Persons of the Trinity as one substance. But they are NOT skeptical, and they CANNOT be skeptical in their LIFE track. They are known for their active mission and community support and what undergirds this is their faith in their concept of God.

Profile of a Deist

Yes, I can perfectly see how a skepticism bred by seeing too much uncertainties in the theological academic track can leak into the life track. I have been there. One can be paralyzed into taking no action. But taking no action ITSELF is an action. One cannot be neutral. Let's say this academic speculation insinuates so deeply into REAL life that when interfacing with a being that cannot be turned off, that one cannot run away from, one has to shout in one's head: "I refuse to pray, to ask for your help, to serve you, to carry out your will for the world, etc. until I can satisfactorily define you in my mind. In the meantime I just run my life as if your nature does NOT matter." In that case, maybe the best label for that person is that he is a DEIST, a child of the enlightenment philosophy. I think it's safe to say that all Christian denominations say that a Deist is NOT a Christian. They will be more willing to call a Mormon / or a Jehovah Witness a Christian than a Deist.

Dealing with the two tracks as a Christian

(to be continued)

Another way to answer the question

@NigelJ and @Lesley comments point us to the right direction. My longer answer attempts to show that the label doesn't exist because:

1. The question itself runs counter to the most fundamental logic in how Christians approach the question "who is God".
2. Language such as "insufficient burden of proof" causing one to "withhold judgment about God's nature" betrays the questioner's holding an enlightenment philosophy of thinking that eventually one can have certain knowledge about God's nature according to the dictates of reason alone.
3. The question blithely disregards the purpose of why Christians hold a Trinitarian proposition as a given, which is still a rational act even though it's a given! Understanding the purpose makes one understand why one cannot be a Christian in that state of mind, and thus the question itself becomes irrelevant.

(I decide not to elaborate on this answer, unless there is interest)