It looks like your Logitech's have the audio signals routed through the center channel, which also controls the volume levels. You could always keep the center channel in the loop (as it apparently makes no noise anyway) and just use it as a volume control for the sub. Worth trying out, at least.
I had an Altec Lansing 4.1 system, and I wanted to salvage the sub for use with a set of Logitech speakers because that sub sucked...not the same one as yours, btw. The volume of the Altec sub was controlled by the right front speaker, but I didn't want to use it that way. So, I modified the two subs so that the amp in the Logitech sub would amplify the woofer in the Altec sub (so the volume of the Altec sub was controlled by the Logitech controls). Worked great.