Am I better to create a new Limited Company at all for the project?
Probably not at this stage. Doing that will immediately incur the costs of a second company (mainly accountancy and insurance and banking). Better to treat it as a "Brand" and make sure all bank transactions are correctly reconciled. That way all your profits/costs will get correctly accounted for.
Other than your project suddenly getting wildly profitable, the only reason I can think of for doing this (creating a new entity) is if you have partners or investors.
Should I utilise my current Limited Company structure for generating
invoices and charging VAT...
Yes, but make sure your company name and address are on the invoices somewhere. If your product is known as "Andy's Sports System", this detail can large and at the top, with the actual company details in a small font at the bottom.
...and potentially move the project to another company at a later
Yes. Worry about making the business idea work. If it does, your accountant can do this for you.
The concerns/queries I have are around is it best to decouple the IT
consultancy business from the project business from day one.
Yes...But it's a trade off against the costs and inconvenience of doing so (see above).
The main reason to create a separation like this is if you plan to sell it, either by attracting partners or investors or as a going concern. Other, larger companies will do this to prevent losses in one company from taking down everything else.
Will this be seen as VAT avoidance in the eyes of HMRC?
Only if the amounts are significant. And if they are, it's because your new side project is doing really well! Congratulations - now pay your accountant to answer this question.
Or make your prices inclusive of VAT, which they should be anyway if you're dealing with consumers.
HMRC = UK tax authority
VAT = "Value Added Tax", i.e. a sales tax