As someone who has paid for a new leachfield, although not from screenprinting, I'd take every precaution to keep that system clear. When I was looking for a space to move my shop, I came in contact with the local arm of the EPA, which informed me that nothing was to be put down a drain connected to a septic system that was not meant for household use. They said I could not connect any kind of "service sink" to a septic system, regardless of how careful I might be, to keep the chemistry out of the groundwater. Your locality may vary. I recall a post somewhere by someone whose "governing authority" permitted them to output into a ditch.
If it were me, aside from assiduously cleaning the ink out of my screens and stuffing my sink drain holes with filter media (which I do anyway) I'd put a tub under my sink as a "settling tank" with a cheap submersible pump raised up off the bottom to allow as much of the "solids" to settle before pumping to my septic system, then get my septic tank cleaned every year or so depending on usage. Alternately, buy one of those not-inexpensive filtering systems. And only use biodegradable chemistry, regardless. Even at the shop I leased which was on a sewer, they allowed me to put in a washout sink in one of the bathrooms, but cautioned me that plumbing problems would be on my dime.
