I believe excommunication is for those who profess faith and are in unrepentant grievous sin. Why would a person who is in the example above even want to fellowship with believers when they can stay home on Sunday morning and watch Meet The Press.
Wouldn't you first nicely ask him simply to withdraw his membership? Or if he's made it clear he has no intention to be part of the body anymore, just take his name off the rolls? The matter can be handled "without process," as we say in Presbyterian circles. No need for formal charges.
Don't have to ask him to withdraw. You should admonish him to repent. If he refuses after a few attempts in person and in writing. Then remove his name from the membership roll. He will thus be part of no congregation and thus be 'excommunicated'.
If a person who has been in membership for sometime announces he is a false convert, I would ask if there is any pastoral issue to be dealt with. What brings him to this conclusion? Sin? Depression? Discouragement? Honest doubts? If he really is a false convert then how can he with any integrity remain in membership?
Excommunication occurs, properly, when the judicatory having jurisdiction (and any and all appeals to broader/higher judicatories have been denied ) determines that someone is persistently and impenitently living in sin.
Having said that, let's divide the question since it is not customary that someone both "renounces their faith" and, at the same time, "claims to be a false convert." Let's take the last first. If someone "claims to be a false convert," that one needs to be earnestly dealt with by the elders. It might start, and probably should, with individual personal counsel by the pastor. Perhaps they have done so because they lack assurance. This needs properly to be addressed. If they maintain such after repeated pastoral counsel, then the judicatory as a whole should join in. All of this should be handled carefully and lovingly and it makes a difference whether the person in view is doing this publicly or not.
If someone renounces their faith, that one should likewise be earnestly and personally engaged. What is the reason for such? Are they willing to talk about it? Can they be recovered? Every effort should be made in this regard, beginning with personal counsel and adding in others where and as necessary.
In all these cases, if and when formal censure comes into view, it ought to proceed step by step, beginning perhaps with admonition/rebuke and then moving to suspension. This assumes that the one in question is not publicly agitating. That creates a different dynamic and impacts the censure. The judicatory must determine what is best for the reclamation of the offender, the purity of the body, and the glory of Christ.
It could be that someone takes such a position and the judicatory, after much dealing, handles this a case without full process. But in no case should earnest dealing with the one renouncing faith or questioning the genuineness of his conversion be short-circuited. The purpose of church discipline is to seek personally to engage one caught in sin.
All this is to say that no simple rule can be laid down in answering the original post. This is not supposed to be easy. It's messy and the pastor(s) and elders must do their jobs in dealing with such a one as much as they can. At the end of the day, if one is persistent in his impenitence, then, of course, excommunication (removal with or without full process) must occur.
But when one does what the original post posits, that should be the beginning, not the end, of a process of engagement with the elders. Even if the person initially refuses to meet with the pastor and/or elders, unless there's a public/scandalous element, patience would suggest that the judicatory not allow itself hastily or unwisely to act but to seek to bring the sinner to repentance. It should be seen as a time for prayer, patience, and repeated attempts to bring the party to repentance.