I agree with CEL. For the CHS section, it is certainly possible to figure out the shear center. It just may not be a profitable endeavor unless your doing them en mass.
To work out the exact-ish shear centre for the CHS, you'd need to resort to fundamental mechanics of materials stuff. Something like:
1) Apply a dummy transverse load.
2) Work out your shear forces and shear flows.
3) Calculate the cross section torsion due to the shear flows.
4) Divide the section torsion by the applied transverse load to find the point at which torsion would be balanced (the shear centre).
With the CHS, the response ought to be dominated by the closed cell shear flow loop through the CHS. As such, an approximate method to find the shear centre would probably just be to ignore the plate altogether. Or at least the outstanding leg of the plate.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.