(A) I think we can avoid this entire problem with a set of different decisions in Manuel I's reign.
But even from a purely geographic perspective, of what practical help Byzantines of that time could be in a proposed Franco-Mongolian conquest of Egypt?
(B) I cannot imagine either the Byzantines or the Crusader States paying tribute to the Mongols in this context. Assuming internal strife within the Mongols' ranks goes similarly to TOTL (the original timeline - is that a valid valid acronym?), will the Mongols be willing to cooperate with the Christian states to divide up the spoils of the Middle East? As far as I am aware, not all of the Crusader principalities involved in cooperating with the Mongols of the Ilkhanate accepted them as their suzerains or paid tribute to them, although I know some of them did.
I'm talking about the pattern which existed during the 1st wave of the Mongolian conquests including those of Batu (who sent requests of submission to the Pope and Emperor Frederic II) and Hulagu. With the passage of time and ongoing disintegration of the Mongolian Empire the pattern had been changing so there could be various models, as for example, between Michael VIII Paleolog and Ilkhanate (married his illegitimate daughter to Ilkhan Abaqa) and then Nogay (to whom he married his another daughter). As I understand, none of these cases involved a formal submission and we are talking about the allied relations. However, situation with Outremer (even if just by the reasons of geography) was different.
Bohemond VI of Antioch, under the influence of his father-in-law
Hetoum I of Armenia, had voluntarily submitted to Mongol authority in 1260, while Hulagu was in power, making Antioch and Tripoli vassal states of the Ilkhanate.
As far as the crusades are involved, during the IX crusade Abaqa sent 10,000 troops to help Edward I and in 1280 Abaqa and
Leo III (king of Armenia) urged the Franks to start a new Crusade but only the Hospitallers and Edward I (who could not come for lack of funds) responded favourably. The rest of Outremer maintained peace with the Mameluks.
'According to the 20th-century historian Runciman, "Had the Mongol alliance been achieved and honestly implemented by the West, the existence of
Outremer would almost certainly have been prolonged. The Mameluks would have been crippled if not destroyed; and the Ilkhanate of Persia would have survived as a power friendly to the Christians and the West"'
https://en.wikipedia.org/wiki/Arghun#Second_mission,_to_Kings_Philip_and_Edward
(C) Why is this a requirement?
Quite obviously, if we are talking strictly about Hulagu, it is rather difficult to be successful when you have two front war on your hands and when a considerable part of your initial force left you (and some of them joined the Mameluks) on the order of their ruler. The conflicts did not end with Berke's and Hulagu's death (in 1271 Abaqa'a main force had been fighting in Turkestan). Anyway, territory of Ilkhanate had limited resources of the nomadic troops comparing to those of the Blue and White Hordes and if they had an alliance instead of confrontation, Ilkhanate could get considerable reinforcements (as was the case with the initial Hulagu's army), which would allow to smash the Mameluks even without too much of a Frankish help.