Lewis,
To answer your question, yes, you can filter your electrolyte solution, but it would be a big hassle. Since there are dissolved and particulate solids that build up in the bath, I think the "operator" should do something to occasionally remove them. My suggestion is to let the liquid sit for a day, which will allow the solids to settle to the bottom of your tank. Then, using a piece of pipe with a few feet of flexible hose attached to it, siphon out a few gallons with the pipe at the bottom so it picks up the settled solids. You can let the solids settle in the collection bucket and return the liquid to the electo tank, or just send it all to the sewer authority. My suggestion is to just get rid of everything you've siphoned out because it will purge both the dissolved and particulate solids out of the electrolyte solution.
Now, for your two birds with one stone question. The "one stone" can be sodium hydroxide (NaOH), but the "two birds" are different concentrations of NaOH - one for electro, one for the lye bath. The electrolyte solution that uses NaOH is much more dilute than the lye bath a piece is soaked in to chemically convert the baked on grease to soap and other crud that can be washed off. You could run electrolysis in a lye bath, but the concentration of NaOH would be too conductive for a 12 volt system.
For what it's worth, I stopped using sodium carbonate a long time ago. Sodium hydroxide keeps the pH a little higher and the anode (I use stainless steel) lasts longer. At the low concentration of NaOH that is needed for electrolysis, I don't consider it being significantly different than sodium carbonate from a safety standpoint. The real safety issue is the higher concentration of sodium hydroxide we use in the lye bath. One final point - neither sodium carbonate or NaOH buffer the solution. The pH you get after dissolving the material is what you get. As the solution gets weaker, the pH will change. I don't think one is less finicky than the other.
Cheryl's lateral is the sewer line from her house to the street. If you have the municipal sewer option, that is the best way to dispose of used cleaning solutions.