TL;DR
Bethane's husband's death, and thus her partial madness, was definitely due to the Deryni persecutions. If he had lived, and she had remained sane, there would have been no "holy shepherdess" for Rimmell to turn to.
After thinking this over for a while and discussing it a bit, I have a suggestion along different lines -- the answer may lie in Bethane's past.
The story, "Bethane", in The Deryni Archives, has a flashback about her life as a young woman.
She was seventeen and expecting her first. Her husband (Darrell) was Deryni. Deryni had been teaching Deryni children secretly, but were betrayed. The children were to be burned. Lord Barrett De Laney exposed himself as Deryni and offered himself in exchange for the children. His offer was accepted. He was blinded with a hot poker. Then Bethane's husband rescued him, but was himself killed (shot with arrows). The child was stillborn. The aftermath, described a bit later in the story:
And their own child, stillborn in the awful after-anguish following
Darrell's death; and then, a long, long time that she lay sick and
despondent at Saint Luke's, not caring if she lived or died, and
something had snapped inside, never to be mended...
Her husband's death, and thus her partial madness, was definitely due to the Deryni persecutions. If Darrell had lived, and she had remained sane, there would have been no "holy shepherdess" for Rimmell to turn to.
On the other hand, Alaric did not know this whole backstory. He did know, however, that Darrell had been Deryni and died trying to save someone else Deryni, IIRC, in the short exchange between the child Alaric and Bethane (from the same story):
"I'm sorry, Darrell," she sobbed, crushing her lover's ring against
her lips. "I'm sorry. Oh, forgive me, my love. Please forgive me, my
love, my life . . . "
<I've left out a bit of description but no dialogue>
"You know what I am, don't you?" he asked, his voice hardly more than
a whisper.
At her not, the grey eyes shuttered for an instant, then turned back
on her again.
"This Darrell--was he killed by a Deryni?"
She shook her head, stifling a sob. "No," she whispered. "He was
Deryni, and died to save another of his kind."
"I think I understand," the boy replied, with a preternaturally wise
nod.
Note that Alaric was eight at the time of this story; it was many years before Deryni Checkmate.
To counter the suggestion that the answer is that Bethane is Deryni, it is absolutely not clear that Bethane was Deryni. The author implies not, in a chat in 2001 logged on this forum
Here is the relevant excerpt ("KK" is the author.
Emphasis mine):
[18:00] Also, Bethany's charm was not purely Deryni--so Bronwyn didn't recognize her danger until it was too late.
[18:00] I think she was just taken by surprise, and acted
without thinking by screaming for Kevin....
[18:00] Can't answer that yet, Kiri.
[18:01] *** Golux Quit (Connection reset by peer)
[18:01] <Kiri`> i'm really wondering if Kennth approved..
[18:01] So, is Bethane of some other magical heritage, like Warin de Gray? Maybe from those strange northerners?
As far as I see in the transcript, the question in the last line I quoted, was not answered by the author.
In light of this discussion, Nezz asked the author (Thank you Nezz!). I'm adding here, the information which was added in a comment:
Well, I figured our best bet was to go straight to the source and ask
directly here. Relevant part (edited for clarity): KK, would you
consider answering one of our burning questions about a character? Is
Bethane Deryni, or does she get her magic some other way? I think
she was probably Deryni, or at lest part Deryni. – Nezz Is
there some other way for people to do magic that's not deryni-based?
We know that half breed Deryni that show any power are still
Deryni. So good answer. There are the Border folk
who know warding lore of their own, and gifts like dowsing. Of
their own. Still could be from Deryni bood.
So it is still an open question, even after consulting the author, whether or not Bethane was of Deryni blood.