It is hard to say if they will prosecute or not - it really depends on how their prosecutor views your actions. It is probably too early to get a solicitor involved, I would wait until they respond to your letter.
As to the content of the letter: yes, apologise. Don't bother mentioning your immigration status as it doesn't excuse your actions. Don't mention a specific amount as it can be seen as trying to 'bribe' them - just offer to repay the difference in price of the two travel cards plus appropriate costs.
I understand why you don't want to post in the general forum, but members such as DaveNewcastle have a lot more experience in these matters than I do.
This is not a good situation to be in, and really not very clever. The only good thing is that it wasn't a TfL inspector as they almost certainly would prosecute.
Disclaimer: Everything that follows is personal opinion - I am not a lawyer.
Since you were using a travel card, it doesn't really matter how many days you did or didn't travel, and the fact that you sometimes touched in/out only makes you look more guilty as I'm guessing you only touched in and out when travelling between Z1/2 stations. At the end of the day all they will consider is that you had a Zone 1-2 card (£493 for four months) but should have been using a Z1-4 card (£840).
That means if nothing else they are due £375 - this is the starting point of any out of court settlement. Add to that their direct costs (say £200) and an amount to keep it out of courts (maybe another £100) you're looking at around £700 for a settlement.
You might want to post a thread to get advice from other forum members.