This is what the stacktrace looked like...
That indicated that this had something to do with Array bindings, which I knew were used quite heavily in this case. However that wasn't the real issue. The earlier version of WebLogic we had didn't use JDBC drivers that made use of DBMS_PICKLER (11gR2 onwards), but WebLogic 12.2 has JDBC drivers that certainly do make use of it.
What the package does it no really relevant, it's an internal package that's to do with array and custom type binding. What is relevant is that it's owned by SYS. And JDBC appears to call the procedure as if it was in the SYS schema.
The trouble was...we had a table with the name 'SYS' that was inside the schema that the application was accessing. This table was created as part of some patching bug, but wasn't rolled back, so once we moved to new JDBC drivers, it all broke.
The solution was to rename that table to something other than SYS and everything went back to normal!