WRT to cleaning the Tiger-cloth - the only thing I do is once dry - I shake it out. Used only during to assist with drying the record after DIW rinsing, the Tiger-Cloth only sees DIW. So, after use, I hang to dry and then shake it out and store in clean bag. The cloth is a very fine weave and anti-static, so it does not hold on to particles/lint - ergo a good snap-shake dislodges any lint/particles, and this has been verified with the UV light (the cloth does not fluoresce). I have been using the same Tiger-cloth for >2yrs. Otherwise, I periodically inspect the cloth with the UV light and someday when it shows that its 'dirty' I will just toss and use a new one; they are not expensive. To me, it's not worth the effort trying to clean it and then use multiple DIW to rinse out the cleaner which if you do not, when you use the cloth to assist with drying the record, it will leave a very slight haze which is the diluted cleaner residue, which I have seen with those microfiber cloths with some nap.
Note that I use a PVA sponge (as discussed in Chapter V of the book) to remove the bulk of the DIW; using the Tiger-cloth only for the final-dry which is not intended to fully dry the record, but to leave a thin-film that prevents any static being developed and then the record air dries in 3-5 minutes.