Thank you Jim; tenacity is not yet my middle name ... but i am working on it
Yes, Beraiz is very determinate in that the term Bilbo originates in the city of Bilbao; actually the city name is spelt Bilbo in the local Basque (Euskera) language.
He is also consistent with other sources in that such naming of the sword is of British attribution, since there was intense trade between the Basques and the British Islands, being a fact that such sword hilt in local terminology was more connected to its shape, namely Bivalve, Conchas (shells), Venera (scallop) and even Medio caracolillo (half snail). Eventually this type of guard was also produced by Portuguese smiths, but note, there was a significant number of 'Biscainho' ( from Biscaia province, capital Bilbao) armourers labouring in Portugal by that time, as recorded.
But while this local terminology came with the later appearing of the typical guard/hilt, the Bilbo term was already in use with the British for the whole sword, such as mentioned by Shkespeare in his work.
Still the Bilbo sword/hilt should not be connected with the Bilbo shackle; besides Beraiz, we must also consider Oxford dictionaries in that such idea is a mith as, for one, such shackle alias was already in use before Shakespeare. Actually some make a point to use the plural form for the shackle version, that is Bilboes.
Concerning the steel used by Toledan masters for their blades, it is of course beyond any doubt that they favoured that from Mondragon but, it is my impression that, the Bilbo term if ever used in such context would be that of relating the steel provenance, that not calling it as such.