Hi Christopher
You are correct when you say that PIP is disregarded when calculating ESA. PIP does not affect you your benefits ot tax liability, you may receive additional premiums on your benefits depending on your circumstances.
Not sure where you get the £6,000 from, apart from savings. If you have over £6,000 in savings and you are in receipt of a means tested benefit such as income based ESA then for every £250 over the £6,000 you loose £1.00 in tariff income. If you have over £16,000 in savings then you would loose all your means tested benefits.
Gary
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
