Are there any reports which designate Ali ibn Abu Talib as a Khalifa?

Thank you for your question!

There is the famous tradition narrated in relation to the verse of the Qur'an {and warn thy nearest of kin} (Q. 26:214). It is narrated with various wording but the general sense of the report is that the Bani Hashim were invited by the Prophet when the verse was revealed. The Messenger, peace and blessings be upon him and his progeny, then said: ((O Bani ‘Abdul Mutālib, I have come to you with the best of this world and the Hereafter. Allah has ordered me to call you unto Him. Who will be my vizier in this affair? Who will be my brother, appointee, and successor amongst you (khalīfati fi kum)?)) Ali, upon him be peace, then said that he would. After which, the Prophet said: ((O Ali! You are my brother, appointee and successor)).
This tradition is narrated in Zaydi and Sunni sources with authentic chains. The word caliph is explicitly used, and the import of the tradition is explicitly stated insomuch that Abu Lahab, who was present said to Abu Talib: "Your son has become your amir!"

Furthermore, al-Hakim narrated a tradition on the authority of Ibn Abbas in his Al-Mustadrak with the following wording:
((Are you not pleased that you are to me what Aaron was to Moses except that there will be no prophet after me? I should not leave unless you are my successor (khalīfati))). Interestingly enough, I know of no authentic tradition from the Prophet, peace and blessings be upon him and his progeny, in which Abu Bakr, Umar or anyone else is referred to as 'caliph.'

And Allah knows best!

IRS

Thank you for question!

As for the authenticity of the first report, I guess that would depend on whose opinion you accept as authoritative. The opinion of Nasrudeen al-Albani does not constitutes a proof or evidence in the religion. His opinion regarding the authenticity of a report is just that...his opinion.

Abu Balj is accepted as reliable by Abu Dawud, at-Tirmidhi, an-Nisaa`i, Ibn Maaja, al-Haakim (who finds no fault in him and says that Imam Muslim relied on him), Ahmed b. Hanbal, al-Bayhaqi and Ibn Hajar al-Haythami.

As for the reference for the second hadith, it can be found in many books of tafsir, hadith and history. If you desire a Sunni reference, you can refer to Al-Majmu' az-Zawaa`id of Ibn Hajar al-Haythami who reported it and said:

Al-Bazzār narrated it, and his wording, as well as that of Ahmed, is condensed. Imam at-Tabarāni narrated it in his Al-Aswat with condensed wording also. The narrators in Ahmed and al-Bazzār are authentic, with the exception of Sharīk. However, he is reliable.

Hopefully, this helps!

And Allah knows best!

IRS