I remember you bringing those two up at one point on the old site! And I could have sworn you told me this already on there, but I can't find it now that I'm looking (of course, haha) and I can't seem to remember - what were their primary types/what type already is decided?
I think Ice/Dark or Ice/Ghost (black tortoise of the north and winter) and Fire/Psychic (vermilion bird of the south and summer) have potential as types, referencing their respective seasons with Ice and Fire and offering an interesting duality in that each has super effective STAB on the other (Fire beats Ice/Dark or Ice/Ghost; Dark or Ghost beats Fire/Psychic), while avoiding giving the bird the same type as Ho-Oh. As an alternative, there's also Water/Fighting for the tortoise (it's apparently a water elemental as the bird is fire, and it's called the Dark Warrior) - this maintains that balance because Water beats Fire but Psychic beats Fighting, so it could still carry that feeling of duality if you want that!
Another cool approach I like is a common weakness, as long as it ties into their lore. The Regis, for example, are all weak to Fighting! Fighting is one of the most humanlike types all around, and the Regis were sealed away by humans, so it sort of makes sense. They're also all defense-oriented and some of the least combat-prone or oriented Legendary trios in the series. Using a common weakness as basis for their types was a really clever move on Game Freak's part in my opinion. This wasn't done at any other point in the canon games as far as I know, but another example of how it could be used - the Legendary Pokémon mascots in my game don't share any types, but they have a shared weakness to Ice. This is because they represent the flow of energy through the region, and cold is basically just an absence of energy; because their energy is so intertwined with their surroundings and they have a tendency to share energy with their environment or absorb energy from it, they simply can't function well in the cold! So if there's a particular concept that would lend itself to a common weakness for your Legendaries, that could be another way to decide on types that somehow correlate without being exactly the same!
That said, I'm not sure what type you already have for them or if that conflicts with it! XP Let me know what you already had in mind and I can try to help further!
Aside from my specific suggestions and just as a general statement, I would not necessarily forbid but definitely discourage giving them both a subtype of Dragon; it's just too common among Legendaries! It's been the secondary type of at least one member of the primary trio for four Generations (and might become Necrozma's second type as well given that it's the disembodied head of a decapitated dragon...), and every member of the trio for two of those, so I think your duo would be better-received if it deviated and was "its own thing!"
Plus, it's always more interesting when they have different types from one another instead of making them match! Solgaleo and Lunala had an excuse to have Psychic in common for the obvious story reasons, but in most cases, I think the trios without shared types are more interesting than the ones with. I loved the Mortality Trio because of the fact that they're tied together by subtle core elements (X, Y and Z shapes and basis on Norse mythology) but look like totally different creatures and don't purely exist as part of a set.
The black tortoise and vermilion bird are conveniently already counterparts, so I don't think you really need to focus on bringing them together by introducing shared elements - I would say you can keep driving them apart as much as you need so long as their core theme (Chinese constellations and seasons) remains strong between them, just as Norse mythology and letters were enough to tie Xerneas and Yveltal together without them sharing common design traits all around!
I'm very interested to see what designs you come up with for these two! C: Good luck!!
Edit: Oh, I totally missed the question about your villains' motives!! Some thoughts on that:
The Vermilion Bird is associated with the Chinese Empress and the Black Tortoise with longevity, so the most obvious goal would be a grab for power (being accepted by the Bird and gaining a following...?) or immortality. But those are definitely too close to N and Lysandre, respectively. XP So I guess not that.
As a better alternative, the Black Tortoise is said to have originated when a king named Xuanwu, seeking enlightenment to become a god, cleansed himself of [either his human flesh or of sin], but the discarded [flesh or sin] took shape as the Black Tortoise demon, which he then defeated. By contrast, the Vermilion Bird is elegant and wise; perhaps your antagonist was seeking to be enlightened or find wisdom (considering their technology/artifice theme, you could reference with alchemy or seeking the wisdom to create the elixir of life?) and seeks the Vermilion Bird for aid and/or accidentally unleashes the Black Tortoise in the process? This could be simplified a little to just "their goal is to vanquish the Tortoise and seek the wisdom of the Bird," with good intentions (betterment of humanity with advanced technology, elixir of life, etc.) but dangerous or downright evil methods (possibly coming back to bite them later, with the Tortoise being a manifestation of imperfection and sin), just as N and Lysandre had good intentions but circumstances led them on the wrong path to achieving those intentions.
Hot and cold also bring to mind tempered steel, which is, in general, another related aspect of the creation and longevity of artificial creations. (You could make them Fire/Steel and Ice/Steel if you don't want to go with separate types for them both!) I totally missed the tech theme of the villains; that would actually work really well as an inspiration for the duo.