Is Telemachus an alpha or an omega in the au? How does that affect the suitors situation?
First, I just wanna say it's so funny that the concept of beta!Telemachus isn't even suggested. When I've written omegaverse in the past, it's always AUs where characters are alphas or omegas. I don't tend to write betas at all bc in many stories I've read, only the "unimportant" side characters tend to be designated as betas and that's not fun to me.
Part of me thinks O!Telemachus is a really interesting idea. He'd be so much like his father that it hurts.
But I would rather imagine him as an alpha. Here's why:
If he were an omega, I don't think the suitors would give him a second thought or take him seriously at all. Maybe a few would try to court him, thinking that if they can't have the top spot, coming in second by winning the prince is a decent consolation prize. But that's the thing- all those suitors are there for Penelope's hand so they can be named king and get all the absolute power that comes with such a title. Why fight for second place? To them, it's go big or go home.
Alpha Telemachus, on the other hand, I think would pose all sorts of problems for the house.
For one, even if he's untrained and a little immature despite being an adult (so like canon), on some level he's like the suitors. He's one of them. An unmated, untamed alpha. And that gives the suitors reason to notice him.
Telemachus would, of course, still try to defend his mother's honor. He hears all the derogatory things the suitors whisper when his mother isn't around and it makes him sick with rage. His alpha instincts tell him to fight back and defend his pack/family.
And he does pick fights. Lots of them. Being untrained, he loses most of them at first but slowly learns a few tricks and moves that let him win here and there. (He never tells Penelope that he's gotten into another fight because he doesn't want her to worry over him. He's not a child anymore and he'll do his part to protect their house!)
And while the suitors think it's fun at first to knock him down, it's a little disconcerting how often Telemachus picks himself back up, bares his teeth, and tries again.
Thankfully, what few wins he earns aren't enough for the suitors to take him super seriously, but Telemachus is a stubborn, stubborn alpha just like his mother. It doesn't matter how often or how badly he's hurt. He won't let any sort of slander against his family stand.
(I've written a whole other post about how a!Penelope would deal w/ the suitors that I'll put up as a companion piece to this ask- long story short, Penelope has to defend her throne and home through trials by combat very frequently.)
So yeah, I like alpha!Telemachus a lot because I think he'd be much more willing to put himself in dangerous situations for his family. He ultimately just wants to protect his sire and to follow in the footsteps of his legendary dam.
As I see it, canon Telemachus is a lot more omega coded and damn, I would actually love to write more about this scrappy little omega picking fights with alphas twice his size bc that's essentially what omega Ody did w/ the massive boar.
But also, it hits so damn good to imagine omega Odysseus home at last with his alpha wife and alpha son, both of whom are taller than him, but love and care for him so fucking much. (Also, alpha Telemachus would act like such a puppy around his omega parent. He'd follow Odysseus on his heels, hang onto every word, and think that his dam was just the coolest person ever. He'd be such an unbelievable mama's boy.)