Like you've said, Hashirama is no idiot when it comes to war. The case is exactly same for Madara. Madara was a genius at manipulation, tactics, and logical thinking.
Senjutsu chakra is highly potent if a user doesn't know how to handle. But this is Madara we're talking about and not an inexperienced Genin.
Senjutsu practitioners learn to draw the energy of nature inside of them, blending it with their own chakra.
Since Senjutsu chakra is therefore mixed with the user's own chakra. So there is no way to separate it and/or hide/seal it away within a person's body.
And moreover, Madara had Hashirama's cells, welcoming the Senjutsu chakra into his body. Which explains how it readily accepted his body. Moreover, Madara was able to decode it within seconds, and figure out the working of it, hence the comment about belittling Senjutsu chakra.
Even if Hashirama tried to fool Madara, I'm sure he would have realized it within moments. But there was no mentioning about it by either of them.