Use Wien's displacement law - as you suggest.
Let's assume that the spectrum you have been given incorporates almost all the flux from the star. This might be ok, so long as the flux is heading towards small numbers at each end of the spectrum? If so, then you can use the temperature-independent shape of a blackbody function to argue that some fixed fraction of the flux is contained below a wavelength which depends only on the temperature. This is a more robust version of Wien's law.
For example 50% of the flux should lie below a wavelength of $4107/T$ $\mu$m.
Thus you sum up your 20 points, using Simpson's rule or Trapezium rule if they are unevenly spaced or have gaps. Then you work out how much flux there is up to the wavelength of each of the wavelength points in turn. When you reach the one where 50% of the total flux is below it, then the temperature is $4107/\lambda$, where $\lambda$ is in $\mu$m.
In principle method 2. is more robust, because you are always trying to find the effective temperature - which is the temperature of a blackbody that gives the same observed integrated flux. Since method 2 is an integral method, it is less sensitive to the detailed shape of the spectrum. Which is good, because stars are not blackbodies.
In contrast, method 1 is really at the mercy of how close to a blackbody the spectrum is.
A possible problem with method 2. though, is that your spectrum does not contain all of the flux. In which case your temperature will be an overestimate (assuming the missing flux is longward of 1000 nm).