No! Don't do it!

pH-Up and similar brands are 98% pure sodium carbonate. It is chlorine-free. Your Aqua-chem is a home chlorination system. It is very useful for killing microbes, and might possibly bleach fabric, but it will NOT substitute for soda ash in tie-dyeing.

Also, if you are going to be squirting your dye onto your tied clothing with plastic bottles, please do NOT use all-purpose dye, such as Rit. I get a lot of sad emails from people who made that mistake. You can use all-purpose dye only in very hot dyebaths. It's no good for direct dye application. The dye to use, instead, is fiber reactive dye, such as Procion MX dyes. See my Sources for Supplies page, which lists many different companies which sell this type of dye.