Interesting question.... I have only used a Thematic Framework ( will attach University of Michigan Study). This one is slightly dated (2010) but was working with two authors new to the topic of Integration... they encouraged me to try the Thematic approach to this 4th leg of my research.... In retrospect it was the best Lens for this piece of research.... Others may be more familiar with Framework Analysis... but on the surface seems like you might be able to go deeper into your analysis with one approach versus trying to combine them in one study... but hopefully others may have different experiences about combining these two lenses to understand your data... Good Luck!
I do think of these as two separate approaches to analysis, where Braun & Clarke's (2006) version of thematic analysis is much more widely used. Indeed, I would say that Rtichie & Lewis' framework analysis never really caught on outside of Britain. But since you are in Britain, you might well have advisors who could guide you in that approach.