For me it all depends what's in the gutters if I use the drum filter or not.
When vacuuming wet sludge I tend use the wet vac without the drum filter.
I've found when dealing with gutter sludge the gutter vac nozzle, bend, pole sections, hoses etc starts to become heavy to use as the crud starts sticking to the inside surfaces and builds up quite quickly. Then all I do is have a bucket of water handy and just dip the nozzle with the pole section and hose attached to the vac. Then suck clean water through it as I work then empty the vac if/when need be. Likewise I tend to do this after every job so it's ready for the next one.
If I have dry crud already in the drum I can either empty it out or put a small amount of water in the drum so it becomes damp when wanting to use the vac for sludge removal with the filter removed.
When vacuuming dry crud I tend to put the filter back in to stop it being sucked through the motors and blown out the sides.
Regards to filter cleaning If the drum filter gets clogged I simply use a garden hose to clean it. I do carry spare drum filters on the van as well so don't have to wait until the other dries.
To be honest most wet vacs are not designed for gutter clearing as I’ve found out as have burnt out a few motors in my time.