My recommendation would be a vacuum belt filter, but it depends on how well the solids dewater. You also didn't state the level of solids in the mix. If the solids aren't too high, then an automated tubular filter would work, but it isn't continuous.

Editor's Note: The person who submitted the question added this qualifier after the answer was posted: To answer your question about level of solids: It can go up to 20% by weight of solids as sometimes it contains floor sweepings as well.

Our expert offered this answer:

Since corrosive and high solids, I would suggest an automated tubular
filter or a conventional membrane filter press -- which is good for cake washing
at minimal wash consumption. The PP filter pressplates are generally
good to 90C so cooling would be required (since Kynar plates are too
expensive).

Another option is an automatic pressure filter, which could
handle the temperature since made of metals. All of these devices are
automated batch, but a cake conveyor/delumper can be used to convert
the batch nature of these three devices into a continuous operation.