Much like FC-20 (for J-10) did not work out. FC-1 (for JF-17) also seldom gets any mention.
"FC" (Fighter China) is a brand established by AVIC specifically for export-oriented fighter aircraft. The "J" series, conversely, serves as the designation code for fighters intended for the PLA's own domestic use.
In the early days, AVIC and the PLA were virtually a single, unified entity. Consequently, AVIC could directly utilize the "J" series codes for the fighter jets it exported—examples include the J-6 and J-7.
Subsequently, China implemented reforms to this system, resulting in a complete separation between state-owned enterprises and the military. AVIC could no longer arbitrarily apply "J" codes to its export fighter models unless it obtained explicit authorization from the PLA. This situation created difficulties for AVIC's export operations. As a result, AVIC began introducing the "FC" brand.
However, lacking the established brand recognition and influence of the PLA's domestic "J" series, market promotion for the "FC" brand proved to be an arduous task. Consequently, following multiple rounds of consultations among various stakeholders, AVIC began experimenting with a new approach: appending an "E" suffix to the PLA's domestic "J-xx" fighter codes to serve as the designation for export variants. This is the origin of the designation you are now familiar with: the J-10CE.
Currently, China effectively operates under a dual-code system where both sets of designations coexist. However, the distinction between these two systems is purely nominal; both ultimately refer to the exact same product.
FC-20: This designation has largely been abandoned.
FC-1: This designation remains in use to this day. However, its application is now confined primarily to internal production and manufacturing systems. Even on the most recently delivered JF-17B3 fighters, one can still observe the "FC1"-prefixed component numbering system within the aircraft's airframe parts inventory.
FC-31 / J-35: Even after the public unveiling of the PLA's domestic J-35 series fighter—the variant intended for the military's own use—CATIC has continued to employ "FC-31" as the designation for the export version. Nevertheless, the physical models of the FC-31 displayed publicly now feature an external configuration that is completely identical to that of the J-35 series fighters, diverging significantly from the appearance of the earlier FC-31 prototypes.
As to whether CATIC will eventually transition to using designations such as J-35E or J-35AE for its export models, we must await further official announcements from the company. For the time being, however, they continue to utilize the "FC-31" designation.