WI: no Gothic war, a Byzantine-Ostrogothic alliance instead

   The problem here would be you need a considerably different Justinian.  The Gothic kingdom, which also included a good chunk of the NW Balkans initially were technically subjects of the empire already, although in reality they had a lot of actual independent.  However Justinian was determined to reunite the empire under central control, i.e. under his rule.  

  Also there was a religious aspect as Justinian and the ruling elite in the east were Orthodox Christians while the bulk of the Goths were following the Arian version of Christianity.  The bulk of the people the ruled were more orthodox Christian - there being no clear break between Orthodox and Catholic sects at this stage.  As such you had a German Arian aristocracy ruling an Orthodox population which was unacceptable to an eastern empire that was growing more intolerant as it regained its power.  As such its very difficult to avoid a war with the religious situation and Justinian's desire to restore the power and unity of the empire.

  There is a way of getting a revived western empire based on the Gothic one but it wouldn't fit the scenario you want because it would mean conflict between the two empires.  When Belisarius initially invaded Italy he had early successes in part because he was a brilliant general and gained some popular support but also because there was a succession crisis within the Gothic sate.  In response to this a number of the Goths offered him the Gothic crown which would have made him king but would have meant war with the eastern empire.  Apart from Belisarius's famous loyalty to Justinian he might have had religious doubts as well but he rejected the idea.

  Other points, are that many people argue that if Justinian had sent Belisarius reinforcements when requested, rather than being jealous of his success and fearful of him being disloyal then the Goths could have been defeated much earlier, saving Italy much devastation as well as the empire much time, men and money.  Also Belisarius had already conquered the Vandal kingdom in the Carthage region a year or two before so that was already in imperial hands.

  If the Goths had been defeated or converted into a western empire I doubt that the Franks could have been fully defeated, even assuming that the religious issues had been resolved..  Their too far away  from the core of the empire and its interests.  Possibly, especially under someone with Belisarius's abilities the southern part of Gaul could be retaken - especially as it had only recently been conquered by the Franks and had a different culture, orientated around the Med.  Possibly also all of Visgothic state, another state ruled by an Arian German tribe.  However that would require not only a quick ending to the Gothic war and Justinian trusting Belisarius but also a lot of resources from the eastern empire and no major wars in the east or on the Balkan front to distract attention.  Again how long such an empire could remain under unified control, especially given the religious differences would be an issue.