but quickly realized that the class of variables used in the problem-based approach will not work with ode45 (or any root-finding problems)
We don't know what brought you to that conclusion, but objective functions involving ODEs should be possible in the problem-based framework using fcn2optimexpr. You should be able to do anything in the problem-based approach that you can with the solver-based approach, except when you want to use the SpecifyObjectiveGradient and SpecifyConstraintGradient options of the nonlinear optimization solvers.
There are, however, other pitfalls in optimization problems involving ODEs, which are discussed here,
