Looking for advice on how to handle a low offer.

Background:

Me, Wife(homemaker), 1 year old. My career is important to me and this position is the perfect next step. I cannot justify moving my family and making two house payments (or rent) for a financial gain of $4,100.00(before expense). The board knows that I am capable of the job and I have a great track record in every aspect of my position.

The numbers:

I am a currently a executive director of a facility A with a $250,000 operating budget,  a ~$5,000,000 capital outlay, and 3 employees. My net salary is $43,600 with 3 weeks PTO, no Ins., and no match. Transportation facility B is offering another executive director position with a net salary $47,800 with 2 weeks PTO, Ins.(only me), and 4.6% match. Transportation B has a $3,000,000 operating budget, a ~10,000,000 capital outlay, and 14 employees.

I have reviewed 11 similar positions that have been offered within the last 2 years.

                      low end             high end           mean
Mean        $79,984.75     $107,932.58     $93,958.67
Median     $72,820.50     $107,000.00     $95,250.00

I understand their logic in paying me based on my current earnings. But their offer does not represent the job. I really looking for suggestions here.

How should I present my information to the board of facility B?

I am willing to lose the offer based on the current number. What can I do to maintain goodwill if they accept my counter offer?

What should I counter with? Salary? maybe a bonus structured on performance? I am willing to put my money where my mouth is.