Migratingbird, did you have any success with your application?
I recently applied for an LTVP for my girlfriend who was on an LTVP under me when we moved to Singapore 2 years ago but cancelled as she got an S-Pass (that time we simply submitted a notarised Statutory Declaration). She is not working now so we needed a new LTVP…
We submitted as before on the 17th Feb with a statutory declaration, but on the 4th March they wrote to say we now needed our letter to be notarised by the UK Embassy as being equivalent to a common-law marriage in the UK, who gave me the same message as you have received - they cannot do this as common-law marriage is not recognised in the UK (and they don't offer this service anyway in Singapore).
We instead submitted a publicly notarised affidavit as advised by MOM, but got a reply from them saying 'your country does not recognise common-law relationship although you have furnished a notarised affidavit to us', and that our application was rejected.
We have appealed but am not confident it will be approved and are currently having to live in a hotel in JB as my partner is Malaysian and can only spend 60 days out of 90 in Singapore, while our rented apartment in Singapore lies empty and I have to sit in a 1.5 jam every morning and evening...
This new rule change is looks like being a major issue for British citizens who are not married. We have thought about getting married to solve this issue (like you want to do it properly) but have read it may still be rejected as you generally need to married for 2 years...