Sounds like an overreaction, yes. I wouldn't resign - the only reason to do that is to avoid getting sacked but as you are in the luxurious position of having a new job lined up with an employer who knows about this and doesn't mind, there's no need to worry about that.
There will be little point pursuing them in a tribunal should they sack you - unfair dismissal compensation is based on actual financial loss incurred, so as this is likely to be a very small amount only, it's unlikely to be worth the legal fees.
Certainly fight it at the moment though, I agree, defend yourself fully and appeal the decision if it comes to it as well. Bring the letter with you and include that as evidence.
Coatgate I found this you said interesting:
"Common consent among employers is that it is virtually impossible to get rid of someone for gross misconduct, even they are found with their hand in the till. Unless the employer ticks every box, crosses every T and dots every i, any tribunal would find in the employee's favour."
I would actually turn that round and say it's perfectly possible to get rid of most people as long as you 'tick every box' etc. If you get decent advice (and follow it) ticking the boxes isn't actually as difficult as lots of people think.