Shias don't believe that Ali was the Fourth Caliph? Are you sure?
If true, that surprises me. Sure, people will interpret events in all conceivable ways and then some, but the role of Caliph has a very public and very political nature. It is arguably entirely political even.
I don't think that very many Shias doubt that Ali succeeded Umar as Caliph. (Edited to add: but not immediately; the third Caliph was of course Uthman. Again, that is not at all controversial; it just happens that Shias care a lot more about Imams than about Caliphs, and may well believe that there ought not to have been any Caliphs at all)
For that matter, very few people doubt that, period. Even outside Muslim communities. It was a historical event, a very public one, about as well documented as any event could be at the time.
It seems to me that the core matters here are:
1. Whether Ali should have been selected as the first Caliph instead of Abu Bakr.
Sure, most Shias will feel that Ali should have been the first Caliph as opposed to the fourth. But I don't think that they will want to challenge the statement that he eventually did in fact become the fourth Caliph.
2. Whether Imams are a meaningful role deserving of acknowledgement by Muslims of good faith.
As I understand it, Imam is a purely Shia concept. Sunnis don't acknowledge Imams at all. In contrast to Caliphs and Emirs, the very existence of Imams is a matter of belief - Shia belief specifically.
All Imams going back to Ali himself, and all members of the Al-Bayt more generally, are presumably chosen by Allah to serve a role with clear supernatural components. All of the Al-Bayt are either Ali himself, his wife Fatimah, or their descendants.
I am disregarding here the use of the expression "Al-Bayt" by Sunni sources, of course. They may use the expression, but it is clearly with a significantly different, wider yet much less exalted and much more peripheral meaning than that of the Shia.
Sunni speak of Muhammad's direct family generally, including his wives, as being the Al-Bayt - but they don't really expect Imams or even martyrs to arise among them, and they certainly aren't interested in making claims about the descendants of Ali and Fatimah to any extent comparable to Shias'. Shias use a considerably more strict definition because they put their Al-Bayt in a much, much more central role.
There are always exceptions - particularly in contingents of hundreds of milliions of people - but I would expect that few and perhaps no Sunni will even acknowledge that the Al-Bayt exist as such, let alone Imams. And it is unusual indeed to find anyone who believes that Imams exist but also that there may be any who are not either Ali or his direct descendants.
