Is it possible to use bifactor model to predict another construct in SEM? Background info: I've explored the dimensionality of a construct with a four-correlated-factor model, which yielded a good model fit but the factor correlations were too high (all above .70). A bifactor model could better capture the dimensionality of this construct, with one general factor and four uncorrelated specific factors.
Problem: I want to check the relation between this construct and an outcome variable. Given that the discriminant validity of the four-correlated-factor model was inadequate, can I use the bifactor model to represent the construct in SEM? I've never come across a research article that did this and I'm wondering if it is feasible to do this analysis.

You are correct; there are relatively few applications of SEM that include latent variables modeled using the bifactor structure. This being said, a handful of articles have used this approach. For example, Tarescavage, et al. (2015) used the bifactor SEM in the context of constricted validation, so you may find it helpful. I also know for a fact that mplus can estimate bifactor SEM models, and I believe lavaan can as well. So to answer your question, yes, you can use the bifactor model to represent your construct in the context of SEM.
If you do decide to take the bifactor SEM approach, there are a couple of things you should be aware of. First, the bifactor model has a tendency to fit even randomly generated data sets well (e.g., see Bonifay & Cai, 2017 for more info), so you should not be surprised if your bifactor SEM model fits better than competing models. Second, SEM  fit indices (e.g., CFI, TLI, RMSEA, SRMR) are known to be sensitive factors such as the number of items, factor loadings, sample size, and estimation. Given this, and also the fact that bifactor SEM model fit is an area that has received little attention in the methodological literature, I would caution you to be wary when using established cutoffs (e.g., those given by Hu & Bentler, 1999) to interpret how well your bifactor SEM model fits the data.
References
Bonifay, W., & Cai, L. (2017). On the complexity of item response theory models. Multivariate Behavioral Research, 52(4), 465-484.
Hu, L. T., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural equation modeling: a multidisciplinary journal, 6(1), 1-55.
Tarescavage, A. M., Finn, J. A., Marek, R. J., Ben-Porath, Y. S., & van Dulmen, M. H. (2015). Premature termination from psychotherapy and internalizing psychopathology: The role of demoralization. Journal of Affective Disorders, 174, 549-555.
