No offense, but what are we supposed to tell the public that will help keep them safe?
To be honest, the most effective way I can think to combat Fen'Harel's endeavors is to make sure the public doesn't know about it (so they won't panic or try to "help" capture his agents / find his locations and just make things worse or get in the way). And the best way to ensure that elves aren't punished for Fen'Harel's efforts afterwards is to ensure that humans don't know about it.
Personally, I agree with this. By which I mean, this is the method which I would use for various reasons (among them, a preference for working in stealth and silence AND concern for protecting elves against prejudice. If willing, I could also see how the other strategy could help. Both have advantages and disadvantages, though. For example:
Meh, it doesn't matter who is listening, if it's one of Solas', they'd just report something he likely already knew. And I doubt he's worried about what mortals might come up with. I picture him arrogant that way, especially after how Cornflakes turned out.
This also means that Solas now has a better idea of what you're doing and can plan against it. This person who offered to help now knows that you (say) want him to investigate Mysterious Artifact X. They have no idea why. They tell Solas. Solas knows about Mysterious Artifact X, and between that knowledge, knowledge of your Quizzie, and any other information he's got from other sleeper agents working for you, he probably has a good idea what you want with Mysterious Artifact X and can take steps to nullify any advantage you would gain from said Artifact. He can also have that Agent tell you slightly erroneous information that will, if not completely lead you off the trail, at least waylay you.
That will ensure that the elves aren't killed, but not necessarily the "most effective".
Lets try an exercise. This is for everyone to answer. Now, for your Lavellans, if Solas really was the big bad the rest of the elf haters around BSN think he is, and his plan would destroy everything and everyone... And you could slow his efforts by alerting anyone who bothers to listen to be wary of elven servants because they could be spies... would you? Or would you let him keep his advantage even if it means the world ends?
It's obviously not going to come down to that, and this is a worst case scenario. I'm just interested in seeing if a Lavellan would actually do that. Would be a hard choice, but certainly better than your clans being gone along with the rest of the world. I don't think Dalish care much about city elves anyway.
I love these games!
For my canon Lavellan (and note, I'm answering in character. I could give you a character with a completely different answer and its reasoning, too): No. My Lavellan would not be telling anyone. A few very trusted people, maybe. I would let him keep the slight advantage. It's a chance to bring back Arlathan. It is arrogant to assume that the world as it is now is the way the world should be or that the current population deserves life more than the Elvhen population and spirits. She doesn't know what will happen to the current people, whether they will die forever, some will survive in the new world, they will be 'reformed' as spirits. Either she can change his methods by employing better means (ie: refusing to press an advantage when it can and will directly effect the innocent**) then she will. If those means fail ... well ... quite frankly, Solas deserved to win, and I wish the new world luck. 
**Yes, I can see the apparent hypocrisy in this. But I hold a different view on that morality than you. I think she WOULD be responsible for the wholesale massacre of elves that might result from her direct action and WOULD NOT be responsible for the deaths Solas causes by his direct action.