The problem is the nature of the Catalan Co. As unruly mercenaries, they are not the best instrument for asserting central control, and as it was, they make enemies the very day they set foot in Byzantine territory.
If they were more successful, they would most likely provoke a reaction in Muslim Anatolia, for instance uniting the various beyliks in a common effort to deal with the Catalans.
The Co. managed to recruit Turcoman cavalry in OTL, but this force was built for plunder and raiding, not state building.
I think you are more likely to see a Turco-Byzantine alliance against the Catalans than you are Catalan propping of Byzantine authority.
BTW, there is not enough evidence about the religious balance of Western Anatolia in this period to make any definitive statement about the percentage Muslim, but the region was certainly not overwhelmingly Christian anymore. There were Byzantinish enclaves, notably Philadelphia, but the countryside would have been fairly Islamic by this time - I doubt Christians former more than half of the overall population of the region.
Anyway, Catalan accomplishments in Asia Minor are likely to be ephemeral, although establishing strong coastal positions is not impossible. If I were a Catalan commander interested in a lasting territorial achievement, I'd go for Aegean Islands. That's still not all that secure with Venice and Genoa floating around, but it seems a better bet than something on the mainland.