The big Y interchange with I-2 is just as good a place for I-69C to end as anywhere. It's a pretty significant interchange. I-2 ends at I-69E for the same reason. AFAIK there are no plans to extend I-2 to Padre Island. There might be other highways or even freeways built there in the future, but not I-2.
If someone just absolutely had to extend the I-69C designation to the border the only realistic options would be overlaying I-69C onto I-2 for a few miles West and then down either Spur 115 or FM 396. Both have portions of limited access, but FM 396 is probably the only one that can be fully upgraded to Interstate quality without much difficulty. Even if both routes were upgraded to freeways, complete with direct Y interchanges with I-2, their route numbers would probably never change. I-69C in all likelihood has its Southern terminus permanently affixed to I-2.