The problem is twofold. Firstöly, the Lateran Council's ban had no effect whatsoever, and the Catholic Church did not become Europe's primarey power player by being slow on the uptake. And secondly, the official policy of the church only supports one position on pretty much all weapons, so distinguishing between the permitted and the non-permitted is both theologically and politically problematic.
If the papacvy had allowed itself to be dragged into these debates (which were very much alive at the time, but mostly among professional users of violence), it is likely that they would have adopted the general tenor and condemned weapons that devalued traditional conceptions of valour and threatened established power structures.
I don't think that catapults or siege engineering would be condemned because the papacy was still culturally part of the Mediterranean setup, where this had a long tradition. A North European pope might do it, though.
Incendiary weapons are probably too insignificant a factor to attract papal condemnation at that point, though gunpowder almost certainly would be included if the tradition exists by 1300.
There may be something said about long spears and the general cruelty of targeting horses and slaughtering fallen riders. Horse archery, too, could become officially frowned upon.
Beyond that it is hard to see what would attract papal bans. There wasn't all that much new weapons technology around. IMO the ban on crossbows was a once-off deviation from course that the popes hads the goosd sense not to repeat.