The bank will take what the bank will take. You never get anything you don't ask for.
The tax assessed value is worthless. You should not be using that for anything.
What does your agent suggest? Have they called the listing agent to find out if there are other bidders on the home?
If there's no offer, it won't hurt to throw you offer up there. $240K on a listed price of $300K is only a 20% discount. If the home is worth 400K - the 80K, then you should be getting more than that of a discount.
My guess though is that if it needs 80K worth of work, then you've got a house that's completely uninhabitable for a while. 80K IS A LOT OF WORK!
Hopefully your agent can help - no one else can answer that "WHAT IF" question for you.
Work through a buyer's agent - there is no charge to you - it only makes sense to have someone represent your best interest.