Tracking errors in batch impossible as Dev Console stops logging or breaks with Server error

Another option could be to create a custom "error" object and then in your batch code you can use try / catch to trap errors and to write a row to the table. This lets you capture the error, the stack trace, plus you can add your own custom information such as what record was being processed at the time, which function you are in etc.

Have you tried not using Developer Console, and while logged in as the user setting up Debug Logs via Setup, click Monitoring | Debug Logs or Logs | Debug Logs.

With this you can configure upto 20 debug logs to be captured. The downside is that other Apex / VF activity they perform in the mean time will also be captured. If however its a dedicated user or the job runs at a time the user is not on the system this should give you what you need.

I often have a batch send me an email with all of the pertinent information I want from the batch run. I declare a string variable that I add to whenever I need to, then put that string into the body of the final email.

String debuglog = '';

...
debuglog += '\nReceived list of objects size ' + scope.size();
...
catch (Exception e)
{
    debuglog += '\ncaught exception ' + e.getMessage();
}
//etc.
...

global void finish(Database.BatchableContext BC)
{

String emailMessage = 
    'Your batch job \"Batch_ClassName\" has finished.  <br/>Log of process:<br/>'
    + debuglog;

Messaging.SingleEmailMessage mail = new Messaging.SingleEmailMessage();
String[] toAddresses = new String[] {a.CreatedBy.email};

mail.setToAddresses(toAddresses);
mail.setReplyTo('[email protected]');
mail.setSenderDisplayName('Salesforce Batch'); 
mail.setSubject('Batch_ClassName job completed');
mail.setPlainTextBody(emailMessage);
mail.setHtmlBody(emailMessage);
}

For most purposes a single email is fine. If you're collecting a lot of information, you could send a separate email for each batch execution as well.

The disadvantage is that you need to explicitly echo everything you need out to the reporting log string, which isn't so bad once you get used to it.