Reflected xss in user name thru cookie

Imagine, that we have user A with name - name<script>alert(1)</script> And user B User B request a sim card and the Add authorization to user A (of course this is not the common way to exploit). As a result we have xss thru user name in flash message thru cookie. And (!) we got properly singed cookie with xss payload messages="29972147bc558baf382bbeb0b829d4efec82de2f$[[\"__json_message\"\0540\05425\054\"Authorization will be given to name<script>alert(1)</script> once this user confirms.\"]]"; Path=/