We hope you didn’t feel that you needed to spend $15 to check this.
You’ve basically got it right – the Collectors Edition pets do not behave in WoW Classic the same as they do in Battle for Azeroth. In Battle for Azeroth, pets are (Battle.net) account-wide, so access to them in one of your World of Warcraft accounts grants them to you across all of your World of Warcraft accounts.
In WoW Classic, it works the classic way: it only grants the CE pets to CE World of Warcraft accounts.
If you play WoW Classic on your “time capsule” account, you’ll get the CE pets. If you play WoW Classic on your account you started in Wrath of the Lich King, you won’t get the CE pets.