The whole thing is a bit odd. He won't drink himself and clearly doesn't want the Inquisitor to but agrees that someone should drink from it, if only to ensure that Cory does not. By default that person is Morrigan. If he was afraid the drinker would discover about him, then surely he wouldn't want anyone to drink from it.
Actually all the drinker might discover is that the codex about Fen'Harel was correct and he was more rebel than trickster. There is actually no reason why the voices in the well should know Solas' true identity. If you were to challenge him and say "you're not really a simple apostate are you?" then I think the voices could tell you if he lied in his answer. But I doubt they would whisper "Look he's a god".
It may also be that he realises Morrigan is not someone to blab without good reason and if she did realise who he was, might well chose not to reveal the information or at the very least discuss it with him first. Even if the voices only told her he was lying about his identity, she would be less likely to worry about this than the Inquisitor.
I think the main reason though, is that he doesn't want the Inquisitor to become the effectual slave of the well in view of the state of events and their position of power. Whether or not he knows at this point that Mythal isn't really dead, the idea of putting your will under the control of the well is clearly not a good one. Morrigan can easily be disposed of if she proves a problem - the Inquisitor cannot.