Multivariate correlations can be challenging. There are actually many ways to build them which are not equivalent - and I admit to not understand their different implications very well.
If you have a normal response, you can use + set_rescor(TRUE) which model the residuals as correlated.
You could also introduce a dummy variable as in Model both response and between group correlations - multivariate brms
Also, maybe it would be easier to move your data to a long format and then have:
bf(response ~ TimeVar * group + var2 * group + (group | study participant))
Which should give you (almost) identical model but one that is easier to work with… You could then simply add some correlation structure by replacing TimeVar * group with (TimeVar | group)
Hope that helps at least a bit!