why not do your inspections prior to being under contract?
If you've already been told that you're the highest and best offer, you definitely don't have to let the bank know THAT DAY that their counter offer was accepted/rejected. Do your inspections NOW and then decide whether you want to move forward or not...
either way you're paying for inspections whether you buy the house or not..but this way you're keeping your options open to back out by not waiting until you have fully signed paperwork...
This was what I was going to suggest. When I ran into this situation I received the counter offer/addendum via email. It had the 0 day inspection period and everything. I told the listing agent that I would get that back to them in 48 hours. During that time we had the inspector go in, do the inspection and give us the okay. THEN my buyer signed the addendum/counter and submitted it in. Everyone was happy.