Like Shani says, you can still apply for the 1+3 after doing an MA, and I know people who have been successful doing this. Or, you can make an argument why you dont need to do another, RT recognised, masters - or at least you could as this is what I did but I am third year now so it might have changed. My circumstances were a) relevant work experience b) the masters I did got recognition the year after I did it, based on the units I took c) I signed up to do the core units from an RT recognised masters in the first year of my PhD. so it's not impossible, but not sure what a difference it makes to the way your application is viewed.
Alternatively, are there other sources of PhD funding in your field, other than ESRC?