I trust people realize, one reason the "no establishment" clause of the Bill of Rights found ready acceptance in the early USA was that the various states often (generally, in 1787) did still have established religions of their own--but the denominations differed. Even moderates and conservatives who took for granted the state should establish some religion could foresee (and I think the conflicts had already started in practice) trouble in trying to establish any one religion on the Federal scale.
The logic of republicanism was indeed at work I think in first preventing the Federal government from imposing one national denomination, and then in eroding support for the individual states having their own separate ones.
However, while they lasted, the various establishments of the states that retained them for a time are in fact examples of the sort of thing the OP asks about; they weren't all abolished forthwith.
Massachusetts is interesting; before the Commonwealth finally bowed to the trend of total disestablishment in the 1830s (IIRC) there was a transitional phase where the law continued to specify there would be an established religion--but threw the choice of which denomination that would be open to the voters. (The voters chose the Unitarian Church, which was thus briefly, by the election of the people, the established church!

)
Another obvious possibility for "consulopapism" in the USA would be Deseret/Utah. Of course the very fact that in Utah there was a settlement committed to one religion was the sort of thing that made the rest of the USA very suspicious and insistent that they not put any formal ties or privileges for the Latter-Day Saints church in their constitution or written laws. But perhaps if by the 1850s there were a few more examples of sectarian settlements that proved to be on the whole more or less normal participants in the US political system, perhaps there could have been more acceptance of the idea that the State of Utah (perhaps, under these circumstances, even called Deseret?) could have a formal union of Church and State, subject of course to Supreme Court rulings and Congressional legislation on behalf of the basic rights of non-members in that state?
The logical tendency is of course for the State to defer to the Church in matters of fundamental principle; under that logic the state becomes administrative/executive machinery for implementing Church rulings and the real political power lies within the Church.
Looked at this way, the system of the Soviet Union was much like "consulopapism!" The Communist Party reserved to itself all decision-making regarding the ultimate goals of society and set broad policies; then the Soviet state machinery focused on implementing this vision, whatever the Party machinery said it would be today. Generally speaking, the people who did these two jobs were one and the same; whether they were acting as Party members or as elected or duly appointed agents of the Soviet state depended on whether they were making broad policy decisions or narrow technical ones.
So really this sort of thing tends to be theocratic, with the "democracy" a puppet of the church establishment rather than the other way round.
Now, I think it's clear enough that the established churches of the various colonies that became states in the USA were not on top of things there; rather, the fear among Congregationalist colonies like Massachusetts was that Tories might impose control over their church in the name of submission to the Church of England, but that the "Church of England" would actually be a mere puppet of Parliament and the King. Similarly after the Revolution, I don't think anyone felt the various established churches of the new states held the whip hand; on the contrary one reason to oppose them was that they were seen as corrupted and mere tools of whatever political faction promoted them. That's "consulopapism" I guess, but it's no way to run a church that someone wants to take seriously!
Aside from Deseret, the closest thing I can picture to your specs would be a Catholic state in the Union. Letting Papists have that degree of control in even one state would send the informal Protestant establishment of the 19th Century USA into a panic and a frothing rage of course, but suppose it happened somehow--say some midwestern territory wound up attracting lots of Catholics and not many Protestants, and somehow the Union went along when a new state government established the Roman Catholic Church in that state, with clauses guaranteeing freedom of religion for dissenters and closely limiting the privileges of the Church.
Well, first of all the state government could hardly dictate policy to the Church, which by its own organization is set in Rome. Going on European precedents they could perhaps secure a conceded right from the Church (which was in the 19th century, IIRC, considered a "missionary" branch rather than a regularly established one--perhaps this status would be changed for this one state) to have a say in personnel--who becomes bishop of what diocese and so on.
Second, American Roman Catholics, as much as they were feared and hated by Protestants who believed they'd be anti-democratic stormtroopers of the Pope, were actually a fractious and independent-minded bunch from Rome's point of view (one reason for that "missionary" status being kept imposed!) who would probably, as they do now, interpret the dictates of Rome in a fashion they find more compatible with the general American way of life.
Thus, the Roman establishment and the enthusiastic American faithful might find another reason that the American practice of disestablishment is a good idea--American Catholic politicians, devoted sons of the Church though they consider themselves, might well refuse to ratify into law rules that the Church hierarchy in Europe, accustomed to more authoritarian states with none of this pesky business of tolerance for minorities, take as elementary duties of any faithful Christian whose hands aren't tied by pesky realities like being in a multi-confessional state for instance. Lacking the excuse of disestablishment, the American Catholics might be headed for an unbuffered head-on collision with Rome they could have sidestepped had they been less gung-ho to establish a Catholic state.
Conceivably the mess might develop into a formal schism between Rome and the American Church; then I suppose at last the American schism might come under the control of the state legislature. By then of course between settlement by non-Catholic Americans, falling-away of people raised in the Church but now disaffected from it, and the schisming of the Church itself in the state, the bloc that comprises those still faithful to the local hierarchy might well be a minority and not entitled to control the state machinery anyway, which would imply either their church comes under the control of non-believers, or they sever the ties of that church to the State forthwith!
It does seem to me that on the whole, having a church tied to a democratic state government is a bad idea; either the democracy gets subordinated to the church or the church becomes a political football in democratic politics.