The short answer is that you need to go down the route of your Option 1 and include xsquared as a second endogenous regressor. If you do this, you may need additional instruments. One source of additional instruments would be squares of some of the other exogenous variables. A quite clever idea is suggested by Wooldridge on p. 237. It's similar to your Option 2 but with an important difference: instead of using xhatsquared as a regressor in your second stage equation, use it as an *instrument*, i.e., estimate
ivreg2 y q (x xsquared = z xhatsquared)