Are you represented by a Buyer's Agent? What did they say?
This is what I would tell my buyer. You can counter their counter. Once they have made a counter, the original offer is "off the table". If that is the maximum you are willing to pay, counter back at the original price. If you are willing to pay more, counter back at that price. If you want to retain their offer of paying all the closing costs, you need to include that in your counter offer, because the effect of a counter offer, is that it replaces the prior offer on the table.
If you are not working with an agent, you might want to consider having an attorney look at your documents. You want to make sure that you do not inadvertently lock yourself into something you would later regret. A good Buyer's Agent will help you determine a fair offer amount and then fight for you. I wish you the best of luck in buying this home and Dare to Dream.
Real Estate Consultant
RE/MAX Palos Verdes Realty