Is there any way to enter a hitman giveaway?

You need to delete free chapter 1 from your steam library.

I have chapters 1 and 2 and I was considering deleting the free game, but I'm not sure if we will be able to get it back. I have deleted games in the past that I activated with a key and it seems that the key will always be associated with your account and you can always get the game back, but I'm not sure what happens if you delete a game that was temporarily free on steam and there was no key activated.

Well, as an alternative you can ask cg to delete your SG account and then create a new one using a different Steam account..
Other than that, no, there shouldn't be any method for being able to once again enter for such packages other than by deleting the associated content from your account.

Keep in mind that what's happening is that SG is checking what content you own [excluding DLCs, since it can't get information from Steam for those] and then restricting access to giveaways which only show as containing owned content [again, due to this, anything that reports DLCs will always show up as unowned]. In this case, Episode 1 is configured as a base game [which can be detected] and the Season 1 pass is only reporting Episode 1 (and not any DLCs) and thus SG assumes you have full ownership of the entire season.

In short, the publisher misconfigured the listing data for the package, so the issue can only be fixed by the publisher or by making it so that SG no longer detects the base game [which at the moment can only be done through deleting the content].

I deleted two free episodes from library and joined those GAs successfully. You can easily restore free episodes later if you need it.