I'm now about 4 monthe into my divorce. We've met a number of times and have agreed on contact with the children and on asset split, etc. and everytime she changes her mind about 3-4 days later. THis means that she is refusing to take on any of the bills for the marital home (inc. Mortgage) or let me come off the joint account. I pay her enough every month to cover all the bills, she only has to earn money for living expenses. She does this with ease.
Latest thing is she canceled arrangements for me to have the children because she says I lied to her about where I was one night. I didn't lie but can't prove it, so she is 'punishing' me but not letting me see my children.
Should I just put everything in the hands of the lawyers and brace myself for a fight? She has refused to attend mediation and says I either agree to her demands or we go to court. Or should I continue to seek an amicable resolution?

Sadly; Mediation remans an OPTION, and if she refuses to go - then you have a fight on your hands; and have two alternatives:

Give her what she wants; to avoid the expense;
or
Go through the Court process. The process also expects both sides to be actively trying find resolution; albeit through lawyers.

My offer of a Lump Sum now stands at a third of what he'd originally offered in July 2007. Each time I acept an Offer - he withdraws it and makes a lower one. Nothing is final until there's a court approved Consent Order.

It seems your ex is playing with the system, and maybe just keeps changing her mind to eek what she can get out of it.

Divorce is like gambling. How much are you prepared to wager and lose. The one that plays the Poker face the best; often intimdates the fair-game-p­layer­ away and wins.

Amicable resolution means you both do what you have agreed,
It is intolerable to keep threatening you every time she changes her mind on a whim.
Get legal on her would be my advice
All the best
Pete