Own thoughts: I have only very limited knowledge of the different axiom systems. I know that TG implies ZFC. I know CH is independent of ZFC. I know that in TG strongly inaccessible cardinals exist (whatever that means), so I would assume that large cardinals exist, too. This answer on MO states that CH cannot be proven or disproven by large cardinals alone due to the Levy-Solovay theorem (whatever that is). EDIT: As I just found out, inaccessible cardinals come before large cardinals. Together with the other link, this would probably answer my question affirmative.
That is all I could collect about this topic on MSE and MO. Answers are appreciated, explicit references would be a nice bonus.