I think that is weird that you do not get possession at close, & now they want to add days to that. I know it is "allowed" but the 2 houses I have lived in, I got possession at close. Or, w/in about 2 days. I guess your realtor knows what is common in your area. She will be the best to help decide if the counter offer is reasonable.
Try not to worry. It will all work out in the end. Keep us posted.