Although the fact it didn't play out that way in the books argues against that; on the other hand, we could say that the falsehood about needing men because saidin was involved WAS the means by which the Bowl would tie Rand to the Tower in Exile. Personally, I think the Bowl was a big enough unknown quantity that there was no way to be certain men wouldn't be needed.

But at one point Rand himself as much as says that if the weather doesn't turn soon there won't be anyone left to save, and the Tower in Exiles ability to rectify that was a massive lever to use against him whether or not they actually did or even would. I wouldn't put it past them to say, "We won't use the Bowl till you agree to our terms, however many millions die in the interim. " Perhaps now that they HAVE used it Rand feels some debt there. Without that ridiculous bargain they made with the Sea Folk they could've used it and STILL held it ransom, insisting Rand deal with them lest the Sea Folk depart in a huff when they can't regain the Bowl.