Would it be considered the same as disobeying Muhammad himself? Assuming the caliph was fulfilling his duties.
Or are there grounds to disobey proclamations by either?
Any help or insight with this is much appreciated.
The only verse I know of is this:
"O you who have believed, obey Allah and obey the Messenger and those in authority among you. And if you disagree over anything, refer it to Allah and the Messenger, if you should believe in Allah and the Last Day. That is the best [way] and best in result." (Quran: 4:59)
As you can see, it does not really mention a Caliph, per se, so it could also work for president, government, your parents.
It is worth noting that in the tafseer, this one for example, obedience is not systematic: you only obey them if they ask you what is already asked of you (which seems to make the verse redundant (in that it says "obey Allah and his messenger, and obey all those who ask you to obey the first too").
But if you read the second tafseer, giving the context of this verse, it seems it talks about times of war (as in, obey your military superior) which is evidently necessary to maintain coherence of an army.
But outside of this, I don't think a caliph can order according to his whims and get his wish done. "Clean my car", "kiss my feet", are not, in my opinion, orders that the Quran is asking Muslims to follow simply because they come from a figure of authority. Otherwise, how is that different from slavery ? But if the order is accepted and shared justly amongst all people - "pay taxes", "raise an army", "be hospitable to these guests" - that may be different. I don't think a figure of authority is allowed to single someone out and order him things.
It seems this verse is for battles and hierarchy respect. Other than that, and for what is a matter of religion, Quran and Sunna are what should be followed by Muslims.