By default no, not used by ImageIntegration.
SNRWeight is effectively the weight ImageIntegration would use if you were to set the ImageIntegration option Image Integration > Weights to Noise Evaluation and Image Integration > Scale estimator to Average absolute deviation from the median.
To use the script's assigned weight in ImageIntegration you need to do the following in the script:
Set Output > Approved action to Copy.
Set Output > Weight keyword to some value, for example IWEIGHT.
Set Expressions > Weighting to the weighting expression that you want to use.
Then click the script's Output subframes button, which will copy all of the approved frames, including the weight keyword and value in the copy's FITS header.
Then, in ImageIntegration:
Set Image Integration > Weights to FITS keyword.
Set Image Integration > Weight keyword to the weight keyword you are using, eg IWEIGHT.
As an example weighting expression, you might try the expression SNRWeight / FWHM^2. Frames with (1) higher contrast (scale), (2) lower noise, and (3) lower FWHM will tend to be weighted higher. (The squaring of FWHM is a heuristic that in some sense "matches" the squaring of scale and noise in SNRWeight, the later done for technical optimality reasons.)
Thanks,
Mike