Though against custom, why not use the Warder link to use Compulsion? Black Ajah have been known to use it (Liandrin's weave).
The version of Compulsion which is known even by modern Black Ajah Aes Sedai is but a pale shadow of real Compulsion. It's not powerful enough to force people's wills, but only to make strong suggestions to them. We see this both from Liandrin (she had puzzled out a version of Compulsion before she even came to the Tower, but even the improved version she uses after picking up a few tricks from Moghedien is laughable from the Forsaken's point of view) and from Verin:
Of course, the thing was not truly Compulsion as ancient texts described it. The weaving went with painful slowness, cobbled together as it was, and there was that need for a reason. It helped a great deal if the object of the weave was emotionally vulnerable, but trust was absolutely essential. Even catching someone by surprise did no good if they were suspicious. That fact cut down its usefulness with men considerably; very few men lacked suspicion around Aes Sedai.
-- The Path of Daggers, Prologue (Verin's PoV)
So while Eldrith could have tried to use Compulsion on her Warder, it's unlikely she would have succeeded, given that he was so determined to track her down and kill her. And even if she had succeeded, it would have taken a great deal of effort: the energy required for Compulsion increases with the mental strength of the victim, and Warders tend to be very strong. Recall how difficult it was for Rahvin (even for one of the strongest Forsaken!) to keep Morgase under Compulsion for a prolonged period.
Why not simply let go of the bond? Then the Warder can be killed without the "side effects."
I don't believe this would be possible. If breaking the bond between Aes Sedai and Warder were that easy, people (especially Black Ajah sisters) would have been doing it more. As far as I can remember, the only example we know of a bond being broken was between
Moiraine and Lan,
and that only happened when
Moiraine was trapped in another dimension (and presumed dead for several books).
In fact, Rand even asks Herid Fel about this after he's bonded by Alanna. Although the old philosopher may not be the most reliable source on Aes Sedai secrets such as the Warder bond, he's still very wise and knowledgeable, and worth listening to:
"What do you know about Aes Sedai and Warders, Herid? About the bond?"
"Warders? Bond? As much as anybody not Aes Sedai, I suppose. Which isn't saying much, mind." Herid sucked at his pipe, not seeming to realise it had gone out. "What did you want to know?"
"Can it be broken?"
"Broken? Oh, no. I don't think so. Unless you mean when the Warder or the Aes Sedai dies. That breaks it, I think. I remember hearing something about the bond once, but I can't remember ..."
-- Lord of Chaos, Chapter 18: A Taste of Solitude