Also see
similar thread and this archive for guidance for
returning resident applications.
Two key observations
a) ILR is a purely domestic status within the UK and the person will be treated as a non-EEA national when traveling in the EU (before Brexit).
b) British citizenship carries with it right of abode, the right to live in the UK. ILR, like any other leave, is a grant and can be revoked much more easily than citizenship.
Also, if a person renounces British citizenship and applies for ILR (perhaps under the returning resident system), would the Home Office not look suspiciously at an application of a person who has just renounced his allegiance to the country, but who wishes to stay in it?
And a quick question: why do you wish to renounce British citizenship and still remain here?
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.