They would have probably spoken the local Vulgate version of Latin, a very (very) early form of the Catalan language.

It wouldn't be too different from Latin, really. Romano-Iberian administration remained intact long after the Romano-Iberian military structure disintegrated (such as it existed). In this older answer u/thejukeboxhero goes into more detail:

The process of creating an independent Visigoth kingdom in the Iberian Peninsula was a gradual one. As in other parts of the Roman world, early 'barbarian' authority derived from their status as imperial enforcers in territories the emperor could no longer effectively administrate. Imperial control over these groups was always tenuous at best, and over the course of the fifth century, groups like the Visigoths increasingly assumed de facto control, even if they ruled in the name of an emperor. Gradually the Visigoths shed their imperial obligations, though the accoutrements of imperial power and authority would linger for quite a while, just as it did in the rest of the Roman world (arguably it still does in some respects).

And keep in mind, the Visigoths had little demographic impact on the population of Iberia. From earlier in the same answer:

Visigoth settlement in the peninsula really does not appear to begin in earnest until the late 490s/early 500s, though the matter is certainly up for debate. As others have pointed out, the Visigoths were always a minority group; Roger Collins estimates no more than 30,000 persons during the period 392 to 419, and the more extreme estimates for the same time frame number only 100,000. The groups that entered Spain would not have been drastically larger. Their impact on the genetic makeup of Hispania would have been minimal.