HDRComposition requires registered images, so the first step is using the StarAlignment tool:
As configured above (default parameters), StarAlignment will write the registered images to FITS files with the _r suffix.
Now the next problem with your images, which is quite unusual (at least for me), is the fact that saturated areas have very low values:
This is a problem for HDRComposition because by default the tool assumes that saturated pixels are close to 1.0 (white) on all channels. To solve this problem we can use the HistogramTransformation tool to clip the highlights. I have clipped the red, green and blue channels to 0.17, 0.11 and 0.17, respectively. The ImageContainer tool allows you to perform this action as a single batch operation:
With the above configuration, the clipped files will carry the _h suffix, so your working images carry now the _r_h suffix. These are the files that I have selected on the HDRComposition tool:
As expected, HDRComposition worked without problems:
The generated linear HDR image has an estimated range of about 26.7 bits (some 100,000,000 discrete pixel values).
One thing to point out about your HDR composition is that the two short exposures are almost redundant. I know this because the last two composition masks (on the above screenshot) are almost identical. To cover the whole dynamic range of this object better, you'd need at least an additional exposure between the second and third ones, that is between the 120s and 15s exposures. Anyway, the result is quite good as you can see in the next two screenshots. This is the HDR image shown with an automatic screen stretch:
And this is the image after a nonlinear stretch with HistogramTransformation (by btransferring STF parameters) and dynamic range compression with HDRMultiscaleTransform:
Hope this helps.