Rookie extensions could only be signed between the end of the July moratorium and October 31st of last year. That is, Bledsoe did not sign an extension of his rookie-scale contract, which is the only avenue through which the Designated Player rule works.
This coming offseason, there are several options for the Suns and Bledsoe. Quoting from Larry Coon's FAQ:
Link:
https://www.cbafaq.com/salarycap.htm#Q44
If Bledsoe chooses to negotiate a new contract, he can sign for up to 5 years under the Larry Bird exception. The Suns will not have to name him their Designated Player, as that only applies to extensions of rookie scale contracts. A new contract between the Suns and Bledsoe would not be an extension.