It appears that instead of conducting a one-way ANOVA, you want to conduct a one-way MANOVA, since you are comparing groups on multiple DVs. In this situation, there's no reason Bonferroni correction shouldn't be done for the three comparisons. Just divide your alpha level by 3 for each of the 3 tests. You don't even need to conduct the MANOVA omnibus test.
Another approach, which is valid but rarely used, is to first conduct the MANOVA omnibus test and then divide your alpha by 3-1=2 (i.e. divide by the number of tests minus 1) for the individual tests--but forfeit significance for all 3 tests if the omnibus test isn't significant. In 2-group designs, this approach controls the Type I error rate for any number of DVs (e.g., for 10 DVS, you can use alpha/9 for the individual tests), provided that the significance of all tests is conditional on significance of the omnibus test. [Reference: Frane, A.V., 2015, "Power and Type I error control for univariate comparisons in multivariate two-group designs" Multivariate Behavioral Research, 50]
If you were instead comparing 3 groups on a single DV, then you could still do a Bonferroni correction for the 3 comparisons. Again, you could do this without conducting an omnibus test. Another approach is to first conduct an ANOVA omnibus test and then conduct the individual tests at the unadjusted alpha level--but forfeit significance for all 3 tests if the omnibus test isn't significant. Note that for more than 3 groups, using the unadjusted alpha level wouldn't control the familywise error rate, so this is a special case where you can get by without adjusting.
In short, you can ALWAYS use Bonferroni to control the error rate for multiple tests, so no person (or software) should tell you that you can't. But in many situations there are other options that use smaller reductions (or no reduction at all) in the comparisonwise alpha level.