When was your satnav last updated ? Does the map display show the route over the new bridge ?
The PCN is a typical one for road user charging locations, nothing unusual in it except that it asks you for the crossing toll as well as the penalty charge. A 2018 case at th e Traffic Penalty Tribunal involving the DART charge, (the Dartford crossing at Dartford near London), the adjudicator pointed out that asking for the crossing toll as well as the penalty charge was not permitted in the powers granted to the authority by the regulations.
Whilst you can submit representations on signage, I suspect these will not succeed and you'd have to take them to the Traffic Penalty Tribunal for a decision and risk the full PCN penalty. As I cross this bridge fairly regularly, I can assure you that prominent signs are in place advising that a toll is payable. Would the argument about the content of the PCN succeed ? Well it could do and you would be able to cite the case mentioned above.
So if you want to go for the double-or-quits gamble you will first need to submit representations against the PCN. Only when Halton Council reject those reps can an appeal be registered at the Traffic Penalty Tribunal.