Nick,
Were the anodes bagged on the cells you worked with? If not they are all over the cell and truly are a bear to collect to process. But if you use anode bags, it seems to be contained in the bags. Cleaning the bags can be a chore but if done in a tank of rinse water it all does eventually settle out. If you use anode bags which run the length of the buss, they are easily emptied with a length of PVC pipe connected to a vacuum receiver by simply vacuuming up the sludge routinely. It settles in the solution drawn off and the electrolyte can be decanted shortly back to the bath. If decanting always decant into an anode bag just to be on the safe side, because, as Nick said, they seem to get everywhere and we try to minimize that. There are anode bags with a rubberized boot at the bottom, I believe the trade name is "crap trap", these are excellent for this process where you routinely vacuum out sludges and leave the bags in place.
When I processed anodes made strictly from jewelers sweeps, the slimes were roasted and ball milled then pretreated with Nitric and Distilled water, this took care of separating the silver (and rarely Pd) and the base metals from the slimes and made for easy filtration and the balance were processed in aqua regia without rinsing or incinerating.
When processing the slimes from e-waste it is different because we try to enhance filtration. The slimes are still roasted and milled but they are then treated in Hydrochloric to eliminate any tin, the bane of filtration if processed directly in Aqua Regia. They are then rinsed well and again roasted to eliminate any Hydrochloric, this allows nitric treatment to separate silver and palladium. After a good rinse and filtration, the remaining insolubles are digested in Aqua regia and refined normally.
The roasting and milling makes digestion much quicker and under no circumstances should your slimes be melted first. Melting just slows the process of digestion.