Coke fines should be captured by the HCGO and should not end up in the LCGO that is typically routed to a Diesel hydrotreater. What range of gas oil is routed to this hydrotreater? For the filter fouling rate to increase after the drum switch is unusual for a diesel unit. It is more common for a HCGO hydrotreater.
The problems I have had with LCGO fouling has been oxygen intrusion causing polymerization of the olefins in the LCGO. A very low level of oxygen can catalyze this polymerization. In one case, the source of oxygen was the straight run diesel that was stored in a tank that was vented to the atmosphere. Oxygen diffused into the straight run diesel. When that stream was blended with the LCGO from the Coker, the oxygen catalyzed the polymerization of the olefins as the feed was preheated.
This does not seem to fit your case. If adding LCO from an FCC helps, then a potential cause is entrainment of asphaltenes into the gas oil that you are routing to this hydrotreater. What is the MCR (or CCR) of this feed? Is the feed clear or is it dark? LCGO should not be dark. It will turn dark as it reacts with oxygen, so you need to look at it right after it is sampled. Asphaltenes will eventually blind the feed filter and you will need to disassemble them to remove them as they will not backflush off at some point.