OK, I have the very annoying requirement "Courtier with Pooka deer affinity"

Now, I have not seen a single Pooka with deer affinity for 110 years. I have been inviting Pookas to my court and arranging marriages for them to procreate and hopefully get lucky before I die of boredom.
Now, since this was mostly a "thing you do when there's nothing else more pressing" and they seem to breed extremely slowly, I wasn't paying too much attention to the splat pairings and offspring produced.

Until I changed to the militant version of summer in order to end the struggle (and buy more time) which meant that I could take concubines: "Great" I thought, "let's breed more Pooka personally!"
However, I'm Firstborn Fae Sidhe, and my offspring are all Sidhe.
Every child, no matter if the mother was mortal(3), Fae (Sidhe, 5) or Pooka(3) became Sidhe.

So my question is this: What determines what splat a fae gets born as?
Will Sidhe and Pooka always breed Sidhe, is there a genetic priority order? Is it random?

As I am looking through the marriages I am beginning to worry that it is the splat of the dominant gender in the marriage contract that decides, in which case I may have reduced my chances of getting more Pooka as marrying unwed courtiers to out-of-court Pooka matrilineally if my courtier was female and vice versa has been my go-to method to bring Pooka to my court.
I really hope my suspicion is wrong, or there will be a lot of forced divorces on the horizon.

Question 2: is there a formula for the chances of getting certain Pooka affinities?
At first glance it seems to be more common with parent affinity, but also many examples of other affinities, not related to either parent. And just to throw my theory off, I found 3 Sidhe children born to 2 pooka parents - However, in CK3 who's the actual parent is never entirely sure.
I would really like to maximize my chances of getting a deer withing a few hundred years, or, if I will never have one since there are none ingame ATM, I'd like to know that too, since that means starting over.
And 133 years in, I'd rather not :-)

I am throwing myself at the mercy of these forums, hoping for guidance - And please let me know if I have missed something obvious, i'd rather be called a fool sooner, and educate myself than living in ignorance and realize that i was a fool at a later time.

Help?