"Always faithful, always ready, always lethal," is a good Chapter motto.
I agree, though, this might feel a bit more Astartes like: "Semper Vigilans, Semper Invictus, Semper Fidelis". Ever Vigilant, Ever Invincible, Ever Faithful. Rather than translating Semper as Always, translating it as Ever, as in [For]ever, I think feels a bit more self assured, which an astartes chapter would be.
The US Marine Corps' "Oorah!" is too on-the-nose, in my opinion. Maybe answer "Semper Fi!" with "Orcus!" the Roman god of death, i.e., the Space Hawks swear to faithfully serve the Emperor's will by killing all enemies of His Immortal Majesty?
I also agree. Though if you wanted to give them a bit of an old world feel, the Russian "Urrah!" is both similar and also ancient at the same time. If you want to go roman, you could always have them use the Barritus, which isn't words, but rather a haunting intonation meant to intimidate. It's chilling if you've ever heard it (some movies use it, and it's also done at historical reenactments). Or you could simply have them not use a war cry and instead fight in solemn silence.
"Always faithful, always ready, always lethal," is a good Chapter motto. The US Marine Corps' "Oorah!" is too on-the-nose, in my opinion.
Being pointed out like this I think I would agree with you.
I am looking to create a chapter that shouldn't feel out of place among Rogue Trader as well as "modern" 40K.
That's getting easier to do with the way the games rules and models have been going. Power Armor will almost always feel like an anachronism. It's both modern and ancient. On one hand if you offered Power armor to the 101st Airborne, they'd love it, but 5th SFG would hate it. You can't integrate with populations, and execute stealthy raids wearing clunky power armor. But a supersoldier used to wearing it and with appropriate modifications, it's a different story...
You've got the framework for a decent homebrew chapter down, the information you've given until now is quite solid. I've gotta agree with Bjorn on the 'Oorah' being a bit too obvious, though. My suggestion would be to have an officer shout something like 'Ave Imperator' first, with the line brothers replying with 'Semper Fi'.
Indeed I am looking at the shout-reply dynamic, though I have been contemplating not having a specific battlecry with putting more emphasis on the motto.
Call and response is very common. But if you want to take a cue from the US military, the Warcry is all about motivation, it's almost never heard in battle. Vocalizations are all about effective communication, and there is way too much going on to worry about a war cry. You also don't want to announce your presence before you're ready. So it could be part of some pre-battle ritual, or something like that. You could also get a bit cheeky, and have them shout curses and obscenities (40k themed of course) at their enemies, that would feel especially like the US Marines...