How can I justify doing a second Master's in Mathematics?
(Note that my perspective is US.) I think your first priority should be applying directly for the doctorate. You can simultaneously apply for an MS, of course, but I suspect that you are underestimating your chances for doctoral admission. 

Yes, candidates are expected to have a pretty broad education in math on entry. Yours seems pretty broad if a bit non standard and you are missing a few things, but you won't know until you apply. 

Doctoral studies are almost always about specialization, however. If you intend to work in an area in which you are already prepared, and say so in your application, then the missing bits might be overlooked. Not necessarily so, but you won't know unless you get into the process. 

A second master's will broaden your education a bit. It may make a difference. Or not. Many places (US, at least) will treat the Bachelor's degree as enough. UK and EU may be more strict.