The OP said that he left the bottom hose off. Wouldn't that mean the mouse nest is in the bottom tank and in that case, wouldn't BACK flushing be the wrong way to go? You'd be trying to push the nest through the core.
On the other hand, he said he can see the nest from the top. In THAT case, back flushing would be the way to go
My suggestion - pull as much of the nest out through and flush from the appropriate direction. You might use a dental mirror or fibre optic inspection camera to check progress as you pull out the crud. I doubt any of the chemicals mentioned above will have any affect on seat stuffing. The mouse crap might be though.