The issue is that once the OP established residence outside of the US, they were no longer an LPR. No longer being an LPR means they are no longer eligible for citizenship.
As stated before, filing tax returns and visiting every few months to "top up" residency does not mean you are resident in the US. USCIS is wise to that sort of behaviour; they will see the FLR in the OP's passport and examine the pattern of travel and will be likely to put two and two together, which is why the OP will need an attorney.
You are either resident in one place or the other. The OP can be resident in the UK and therefore eligible for ILR and not for US citizenship, or they can be resident in the US and not eligible for ILR.