Resampling: how many samples to zero-stuff or down sample?
When you up sample by 160, if you wanted to filter twice, you would actually be applying an "anti-image" filter.  The next filter for decimation would be the "anti-alias" filter.  But in you case, you could just use the anti alias filter once and accomplish the same.  I think perhaps you need to consider what zero stuffing means, to put so many zeros between samples and decimate is to throw away so many samples inbetween. 

Btw, you do compute samples for the zeros when you filter.  If you want to understand better, try plotting the spectrum of each stage, and what would happen if only using the 13.

If I understand your question properly, you would effective just upsampling by 13, filter, then downsample.  You'd be back where you started