I've been invited to an Eberron game and have been making a Warforged Psi Warrior, except I'd like to replace the racial trait "Specialized Design" with something that fits the character's actual specialized design by instead allowing them to cast Encode thoughts. Is there something I'm missing which allows the creation of variant racial traits for races which you don't own when you're only doing it within a campaign that does include the content? I understand how content is shared on D&D Beyond, but I'm not looking to make an entire variant race or background or feat, just a variant racial trait for a character I already have made, is this impossible as a player?

My current work-around for this is creating a Warforged racial feat that adds the choice of any 1 cantrip since I don't have the ability to choose encode thoughts as a spell while making the feat but I can choose the spell on the character in the campaign... this seems to be enough for my purposes, but I don't like that I'm just giving myself the feature and abstaining from making a choice for the Specialized Design trait. Apart from having my DM make the variant race for me is there any better way to clean things up?

You can only create homebrew based on content you own or have homebrewed yourself. This includes things like making a variant of a race or adding a spell to a homebrew.

So yeah, you would need to either have your DM homebrew the variant for you, or purchase the race ala carte from the Eberron book. You'd also need to own Encode Thoughts if you don't already, which you can purchase ala carte from the Ravnica book.

I feel like there's enough different ways to do it that that shouldn't be the case... or at least, that there should be the ability to make homebrew specifically within a campaign, like if I wanted to make an custom magic item in game that had a non-SRD spell or something and wanted to show my DM the completed item for approval? 
Additionally, knowing that I shouldn't be able to make homebrew stuff that contains non SRD content, there are still some things that I can do and probably shouldn't but are still fine? For instance I can modify my background to include an expanded spell list that contains just "encode thoughts" as well, but then I would need a feat that allowed me a choice of a cantrip to get it... or as I mentioned before, if I make a feat I can make it a racial feat for a race that I don't own the content for as well? It seems like I'm good at finding loopholes but none of them fit just right. 
For all the Tiefling variants, how do those variant options work? Are they all just written in as options on the main race or is each variant Tiefling it's own race? like, their little groups of spells are so interchangeable, even with elves, I don't see why homebrew racial trait variants are tied to an entire copy of the race, can't there be an "Optional Trait Manager" for race similar to the "Optional Feature Manager" for the feature variants added with Tasha's?

I mean, you can make homebrew for yourself or that you plan to use in a campaign which replicates published content, so long as you don't try to publish it on DDB. That's allowed, and an option for people who own hard copies of books. So since you have Encode Thoughts shared with you, you could look at the spell description carefully, maybe make a homebrew copy of a similar-ish spell from the Basic Rules to use as a template, and then make the relevant changes. Then you'd have your own personal Encode Thoughts spell that you could use however you wish in your own homebrews.

The Tiefling variants for all the different circles of the Hells are variants of the standard Tiefling race, each one with a trait added as a Replacement for the trait that grants their Infernal Legacy spellcasting.