The appraisal really ties your hands on this one.
If the buyer is using FHA, they absolutely will not finance anything above the appraised value. Then, depending on how much $ the buyer has available for a down payment, he will also be limited on how much he can offer.
The absolutely highest LTV any bank will finance is 96.5% through an FHA loan. So, if it appraises for $400k, the most he could finance is $386k. Then, if he only has $12k for a down payment, he may be forced to offer $398 so he could still hit the LTV numbers (hence the $2k under appraised value offer).
I think your best case scenario is to counter with the amount of the appraised value. The only other option would be to hire another appraisal company and hope it comes in higher.
It does put you in a tight spot with your money down situation, but there are options where the bank will finance a higher rate if you pay mortgage insurance. I'm not going to lie, mortgage insurance is an expensive rip-off. You end up spending thousands of dollars up front, then hundreds of dollars each month, just so the bank is protected. I know current FHA insurance rates are to the tune of 1.2% up front and 1.2% per year (depending on the LTV it can be a little higher or lower), which equates to $1200 up front and $100/month for every $100k financed.
I'm not a familiar with non-FHA loans as that's all I've dealt with recently (including my current refinance).