Actually the tax relief a corporate body obtains on its corporate Gift Aid donations to charities, whilst not the key motivation for the gift, is nonetheless important to that corporate donor.
CICs have no special tax reliefs, unlike charities - as you will sadly know by now. This is one of the deterrents to the further growth of the CIC legal form.
For guidance on the Gift Aid regime and how corporate bodies can make donations under it to organisations recognised for tax purposes as charities (its a tax law test), see the guidance notes on HMRC's website:
To what extent you could use a charity alongside the CIC and what the relationship between them might potentially be is imposssible to say without knowing a great deal more about your situation and your ideas for the future. You do need to take specialist charity legal advice and some good tax advice (including on the VAT issues) - otherwise you may find some very unhelpful and even unpleasnat consequences. I'm happy to advise on a professional basis if you wish - do contact me.