What test is appropriate to use and why? If you want to see the relationship between the dvs and ivs and whether your dvs are influenced by the ivs? Can you combined your IVs? or you need to do it in a separate test?
your question is not specific enough for giving a clear answer to that.
MANOVA of course is the tool for analyzing , whether, e.g. a manipulation, has an influence on three DV's which may all measure the same thing, in different flavors, e.g. IV-Training; DV's ability/test scores. However, if the three DV's you got, measure different things, then the MANOVA is not appropriate, and the results are hardly to interpret.
So, if you got a priori hypothesis for why, the IV's should specifically influence - each - DV differently. Multiple testing is totally fine. In this case, a multiple regression would be appropriate (but take care in not interpreting the effect estimate of the binary IV on a 'continuous' scale). If you do not have a priori hypotheses, multiple testing is still fine, however it is by definition an exploration. This means, multiple testing should lead to adjusted significance criteria (e.g. Bonferroni); and furthermore, if you find something, you still need to test it again, i.e. make a prediction from your observation. Optimally, pre-registrater your study. Then you can draw conclusions afterwards.
Hope this helps,
Best René
Ps: if regression sounds good to you, mixed effect regression is even a bigger plus from a statistical point of view. Bayesian methods also.