I love the basis of this theory.
My own head canon angle is that:
Ed's Proximity to Oswald = Safety = Time to Preen
Ed's Distance from Oswald = Chaos = Survival
TLDR - Edward becomes a visual wreck because he's at war with himself. The external reflects the internal.
We do get to see early on in S1 and S2 that Ed and Riddler both have a rather strict styling. They're both clean cut, they're both put together. Ed early on has a very clean appearance that he upkeeps well and we see that when going episode to episode, even when he's working out in the field or in the lab for cases. The lad keeps it clean.
The only time we see his appearance get screwed and relatively stay is when Riddler is screwing with him and forcing him to react in ways that make him have to jump to the next task quickly. Like when he has to hunt down Kristen's pieces, he's a wreck through the whole thing because he can't stop.
When he and Riddler find a way to get along better in the morgue, the next day he's got himself together again. Prim proper, clean cut, everything's aligned again.
As soon as the two start to deviate, it usually becomes one half needing to react quickly and near continuously to the other. Making it so that one half has no real time to care to themselves. Figuring out what's next or needing to assure safety is the biggest thing, there's no time for hair cuts or a new shirt. Riddler threatened to kill Ed's new crush, so now we have to constantly deal with this and provide around-the-clock preventative care.
Which brings on another fun kind of facet.
Ed and Riddler typically find balance more often when Oswald's around.
A lot of Ed's relationships have him pushed to choose to be a version of himself he's irreparably outgrown, a contained gentleness, or the dangerous Riddler. And as we see through the show, neither half really do well when the other is expected to be spotlight 100%.
Meanwhile, Oswald adores Ed's ability to be both gentle and cruel. He appreciates Ed's ability to blend in and be a civilian while concocting a domestic terrorist plot in his head. Oswald likes both sides, benefits from both sides being present and working together.
Even when Oswald's love comes to light, the entirety of Ed doesn't split or find imbalance in those moments. He's not fighting himself in a mirror. It's not Ed vs Riddler. It's Edward vs Oswald.
So, going back.
Oswald is a safety net. Ed typically has fewer splits, fewer breakdowns, and when something goes wrong Oswald is soon there to fix it. He puts resources into it. Especially the romantically inclined Oswald, who would do just about anything for that pretty, pretty man if he asked. The kind of chaos that would normally cause a rift between Ed and Riddler, doesn't happen. They stay functionally as one. There's no need for one or the other to be running constantly attempting to solve the next clue their other half left.
Ed or Riddler can do what they want at their chosen pace. They only really need to take care of Oswald, something they get extremely adept at doing. And even while they are nearly always on the move for something to do with Oswald, his candidacy, or his familial fallouts, his looks stay clean, well kempt, and pristine.
He doesn't see the challenges he accomplishes for Oswald on the same level as his own challenges.
Which makes sense.
Someone else's problems usually feel easier to solve because you aren't in them. You can make choices and find solutions because you have a clearer mind from the situation, you don't need to fuss over every detail.
Oswald is the safety net.
Oswald gives Edward balance in most situations, gives him a safety net, accidentally keeps both halves of Edward in a state where they can exist as one (even if at times one seems to take more of the lead, there isn't a fight for power between them).
When Edward is fighting himself, he's in survival mode.
When you're in survival mode, there's no need to take care of yourself or your appearance. He's trying to make it to the next clue of what's about to happen that his other half set up. And even in the case where Riddler thought Ed was screwing with him, but turned out to be the chip in his head, it was the sake kind of survivalist reaction. There was a puzzle to solve, there was something to chase, there were answers to find, he couldn't stop until it was handled because things were out of their control.
When he's running around, figuring out what happened to Haven? He gets shaggy and run down. He doesn't care to fix himself up, there's no need, he HAS to solve this issue.
Then he finds Oswald.
Edward solves his head issue.
And oop! Literally that next afternoon he's showered, shaved, hair's cut and styled, he's got new clothes, he's domesticated again, and he's completely pieced back together. He no longer looks like a pickle found behind the couch with hair stuck to it. He's got the man by his side again that allows him to relax.
Relatively.
Edward's got that dog in him (read "dog" as 'Tism).
When you're hyperfocused on things, when you're in survival mode, when you believe that something NEEDS to have 100% of your attention there's just no breathing room or time for upkeep. The energy is going where the person believes it's needed most, and usually it's into problem solving the chaos his other half has made and trying to get a step ahead to prevent something terrible from happening.
There's no energy for brushing your teeth every day.
There's no care to change the shirt on it's third day of wearing.
There's no need to brush the hair, bigger things are happening!
He HAS to solve the clues and riddles of what's happening, what the other half has planned, there's just no time to stop. Rest is usually forced when exhaustion wins out over everything else.
Oswald is safety.
Oswald is known chaos.
Oswald is an opportunity to solve someone else's problems.
And Oswald's problems feel more trivial to Edward.
While Oswald will pamper Edward.
And you know that man would tell Ed to his face if his breath stank from forgetting to brush and would ask about why he's such a wreck if his looks started slipping before taking on the task of solving Ed's problems himself.
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