Giveaway creator has full control over who can claim a "bundle" giveaway, for any entrants whom have any component(s) of a bundle (so long as those components are still separately purchasable outside of the package, and thus not exclusive to it- at the point where it's exclusive, you may want to confirm your rights with staff before proceeding).
In other words, if you're giving away a game and a DLC, you can choose to exclude anyone who owns the base game [though you'll always want to make sure to say such in your description as you put the giveaway up], making sure only people who don't already own the base game can enter for the DLC.

Similarly, entrants have the right to enter so long as they don't own any single component of a bundle. You'll typically want to add someone on Steam and then go to the store page for each of the DLCs, where the "Your friends who own this product" section can be used to determine what DLCs they do and don't have. If you want to avoid that stress, it's generally best to restrict giveaways to those who don't already own the base game- though, in the case of situations like these, where the base game has been free and the DLCs are relatively hard to obtain, it's highly appreciated when GA creators are willing to allow people to enter for DLCs alone.

It should be a single-key or gift, so it's extremely likely that that'll be possible.
On the other hand, the base game has not only been free, it's been bundled 10 times over, so even if you can't find someone with a spare copy to offer you now, you'll likely find one in the future. It may be worth considering if the Collector's Bundle is actually the version you want to enter giveaways for. :)