IIRC, Louis XIV had stopped war with the HRE during the 2nd Vienna campaign but this was an extent of "cooperation". Needless to say that the French diplomacy had been doing at that time everything possible to prevent the PLC from joining anti-Ottoman coalition.
In the time in question France is ruled by Louis XI who had close relations with
Francesco I Sforza, the Duke of Milan, and Republic of Venice and bad relations with the House of Savoy (regardless or perhaps because of being connected by marriage

) and
Ferdinand I of Naples who after the fall of Burgundy was trying to improve them. The same goes for the papal States: the relations started improving after the issue of the "Burgundian inheritance" being settled in 1482 by the Treaty of Picquigny. So, a lot depends on the timing of the Ottoman conquest which, IIRC, was not defined in OP. Surely, it does not make too much sense to talk about 1480 (OTL landing in Otranto) because in a year Mehmend was dead. So the time frame should be a conquest of Bosnia and Albania because for landing in Italy a base on the coast of Adriatic is needed. In would also need an earlier end of the war with Venice (in OTL 1463 - 79) or a complete avoidance of it, again, based upon the date of Mehmed's death. So, let's assume that the landing in Southern Italy happens in 1460's - early 1470's.
Prior to 1477 Louis XI is concerned with Charles the Bold still being around and between 1477 and 1482 with settling the issue of the "Burgundian inheritance". His main opponent, Maximillian is busy with the same issue but from an opposite side. I may be wrong but none of them looks like an easily excitable type ready to forget his own interests for the sake of the Papacy, not to mention the Italian states. Louis may be willing to do something to help Milan but, IMO, the chances of the Ottomans going all the way to Lombardy are rather slim even under the most favorable scenario: by going to Italy that early Mehmed leaves unsettled cases elsewhere (and even a conquered Albania proved to be a serious source of trouble) including problems with Karaman and
Akkoyunlu(White Sheep)
Turkmens who were threatening Anatolia so it is quite possible that in a midst of an entertainment he would be forced to drop the offense, leave some garrisons and sail back to deal with the problems on the Balkans and/or Anatolia.
As for the Rome, well, the Popes had been ruling from Avignon for quite a while and, IIRC, it was still a Papal territory so they could get back there (probably Louis would be happy to have them under his control). An idea that a Pope would stay in Rome to get captured by the Ottomans does not look plausible to me.
The rest is an issue of future: if the Ottomans are still keep trying to expand their empire in all possible directions, holding to the Central Italy for a long time may prove difficult even if they are using a model of the vassal states.