Why do you assume the Dornish don’t produce their own silk? If mulberry can grow in India, South Africa, Iran, etc. it can probably grow in Dorne. Given that Dornish silk techniques are distinctive enough to be recognized on sight – “painted, not sewn. The Dornish paint their silks, I’ve heard” – and have their own distinct name, I’d lean towards them producing it.
So I don’t think they’re competitors with the Free Cities, I think they’re business partners. Dorne sells wine, peppers, citrus fruit, and sand-silk and they buy dyes, spices, textiles, etc. from the Free Cities.
As to why the Dornish are the only Westerosi who have a silk industry, my guess would be that, as silk is otherwise found in Essos (it can be bought in Qarth but given the city’s commercial focus is likely a middleman, it’s manufactured in Naath, and given that the Silk Road is one of the roads that connect western Essos with Yi Ti is likely also manufactured there), it was brought over by those Rhoynar artisans we hear of from Nymeria’s Conquest.