Thanks for your question
I am sorry to hear that you are being pursued for a tax credit review.
In answer to your questions
1) Yes they have every right to ask for evidence to back up the child, if a claim for the child element of tax credits was claimed
2) You are mixing up your legislations, HMRC (Taxes) only have the right to ask for personal bank details IF they feel there is a need to do so, however as a sole trader if you used your personal bank account (which is fine, as its only limited companies that have to use a business bank account by law) then both HMRC (Taxes) and HMRC (Tax credits) have the right to ask for evidence of the income that relates to your self employed earnings, as you choose to use one bank account for all purposes. So whilst you may feel you wish to blank out the personal spending, HMRC need to see this was personal (as opposed to business) spending, so I would not blank this out, I would merely highlight it as being personal not business, or highlight the business (as this would be easier. But I am afraid, the moment you chose to use a personal bank account for all, you put yourself in the position to make all available to HMRC.
So therefore the fact this was a also used for business overrides the fact it was primarily a personal account. And it would be advisable to comply.
I am sure your bank account verifies the business income, and any evidence you can submit for your child, will soon get this matter cleared up for you.
Feel free to ask any follow up questions, on the answer I have supplied, but if you could take the trouble to rate the level of service I have provided, it would be very much appreciated, as this ensures I am paid.
I actually noticed that you worked for HMRC hence I felt I had to call a Tax help office to verify your answer to my question and I was advised tthat I could blank out out my personal expenditure as HMRC has no statutory rights to see my personal bankaccount. So here you go!