Ah, Ok.. I also understand this much better now. I think the main problem I have with this, is the total transparency of the transactions. The only tx that will be auditable will be the settlement tx on the Blockchain and the tx when the channel is opened. The 1000s of tx's that are done with the channel is not auditable. 
How will this be handled in a dispute?
How will this be handled in a dispute?
Interesting question. I think asking both parties to share all transaction/HTLC would be fastest way, even though there's anonymity/privacy concern to both parties and anyone who use their channel to route transaction.
No, it is not about anonymity and privacy, remember with on-chain tx's all the tx's are recorded on the Blockchain, so it is easy for someone to audit the tx history. Will the LN add another layer of anonymity to your tx's, acting as some kind of mixer service, where tx's are done off-chain?
How will my auditors be able to audit my books, if I only provided them with a opening tx and a closing tx on the Blockchain?
Sorry i wasn't being specific, what i meant was both parties share all commitment/updated transaction which happen on LN/off-chain. IMO someone can't manipulate the auditor by only show commitment transaction which give him/her advantage since the other would do the opposite or being honest.
But this won't work if both parties cooperate to manipulate the auditor or only one people who want to show the commitment transaction.
Also, as i mentioned before, few users might think about their privacy/anonymity if the auditor decide to use method that i just mentioned. Please CMIIW.