Total Score is defined with the number of those correctly-responded (dichotomous) items;
Sub-score is defined as the total score associated with the sub-scale;
Overall Score is defined as the total score associated with all testing items.
For Total Score, the Overall Score is the summation of its Sub-scores which is called Additivity.
For Item Response Theory (IRT)-ability (theta parameter), the relationship between Overall Score and Sub-scores is unavailable.
Comment: (5) implies IRT has no Additivity. Therefore, with IRT-ability, the sub-scores and Overall Score can not be available simultaneously. This fact strongly indicates that IRT is not a correct theory for high-stake scoring while Total Score in (4) is (although only is as a special case).
Depending on what software you use for analysis, you have to specify relationship(covariance) between latent constructs. Otherwise, IRT's conditional independence assumption may be reason for 'no relationship'.
(1) In MIRT, all the latent variables represent the scores associated with sub-scales, but no latent variable is for Overall scale. Further, the covariance between sub-scales is the linear part of the mutual relations between subscales, the mutual infomation beyond the linear part and those interactions associated with more than two sub-scales are totally missing in MIRT. Again, key argument is that, in MIRT, there is no way to represent the Overall Score.
(2) In multivariate statistics, the ONLY reason to put all the variables into a single system is that they are interactive, otherwise, those (jointly) independent variables should be studied individually (and therefore, easily). In IRT, the assumption of conditional independence shouldn't be there because, in real world, it is rare to be true. Now, issue is that, without any unrealistic precondition, why IRT can not express its OVERALL score in terms of its sub-scores, i.e. Does IRT have its OVERALL score and what latent variable (theta-parameter) in IRT stands for?