I need help dealing with this situation...

Well here's hoping it works out with SG, otherwise you're gonna have to buy it to fix the issue.
Summer sale is just 6 weeks away, so worst case you can pick it up for a few bucks then.

It's gone as low as 80% off so it'll only cost a few euro's at most.

I want to say that DLC in your library never shows up on Steamgifts – which is also the reason why you could technically enter a giveaway for a DLC that you already have, so you have to be extra careful with DLC giveaways!

"There are so many issues with the Steam API not reporting owned DLC that it would cause a lot of issues for almost everyone if those counted."

SGTools is going to operate somewhat differently than SG itself is, due to the fact that SGT checks against the User API, rather than the Developer API (which is why SGT requires you to be logged in to community to use it, unlike SG). So when SGT notes that some DLCs don't show ownership correctly, they're presumably talking more (DLCs linked to other DLCs) or (the standalone expansion configuration a small number of older Steam games have) and similar setups that completely glitch out the API, not the standard DLC ownership detection issue SG encounters.

"4 pieces of DLC show as not activated, but I don't know if that matters for entering sgtools rule protected giveaways."

Going off what I noted above, SGT can freely check against DLC ownership for things like SGT filters. The exceptions it needs to set against that (ie, the number of DLCs with weird quirks) should be relatively few. Thus, working off the fact that it should be a rare occurance, we can determine that the Dishonored DLCs showing up in SGT's Non-Activated detections list is actually likely to be an outcome of the quirk of how this version of Dishonored activated.

(Side-note, knsys (SGT's developer) is extremely friendly and responsive, so anyone looking for clearer or more reliable answers on anything SGT related is recommended to confirm such matters with him directly through the associated thread on SG.)

On the other hand, the fact that it's in the collapsed/hidden list for that section on SGT means that it's already been flagged as an exception on SGT, meaning you're guaranteed not to have any problems with those DLCs on SGT. That is to say, Games and DLCs that haven't been marked as having known issues wouldn't be in the respective collapsed sections, but visible immediately upon loading the page. Those're the ones you'd have to worry about as far as being blocked on SGT filters.