Every time one thinks of something clever one should consider if it fits the principle of least surprise.
Don't do what specifically ? and why not ? and what should I do instead ?
All of these answers are missing in your comment.
WhatsApp is doing something quite similar - displaying a small search icon. When you click on it - the search box expands to 100% (with animation, of course) and the keyboard input pops up.
But for example - on map based apps I would personally prefer to have the map with as little distractions (icons, buttons,text inputs, labels) as possible.
In Google's Android Chrome the url box is hidden once you start scrolling down. I like that . And they probably have the data to show you that other people like that as well, because this 'hide-when-scrolling-down' feature is there at least a year or so, If I remember correctly.
Also, every time (for the first 5 times) that he clicks on the 'submit search' button/icon - make an animation that will minimize it, allowing him to see how it actually becomes smaller and more compact.
Also, when you get to the point of showing only the minimized search box - you can show him an animated tool top that will point to the small button. Again, you can do that only on the first X attempts .
Also, you can track and analyze the 'learning success rate' of your users. If they actually keep forgetting how/where to look for the search button (after they made x successful search attempts) - then YOU are probably right.
But this requires to be more data driven - try something and then measure its ssuccess etc.
After a few attempts you should (hopefully) see one way of doing it that is by far more effective than the others. But again - this requires data analytics.