Is it possible to have a constant store time varing data? Note, I do not want to read this data into the model using FromWorkspace blocks because these blocks are not supported when using Simulink Design Verifier.
NO. Simulink is permitted to reason about the effect that constants have on the output, under the assumption that they are indeed constant. It is permitted to uses them to reason about whether indexes will be in range, about whether derivatives will be singular, reason about whether an integration will overflow, reason about whether a variant system will be selected, and so on. If the constant could change with time, then that would not be possible.
For example, V1+1/R0 is fine when R0 is a constant that can be tested as not being close to realmin(), but if the "constant" were varying, then it would have no idea whether R0 is ever 0 or not.
If you have fixed input "constants" then you can in theory break your model up piecewise in time, verify, and glue the pieces together. Though using a lookup table like Markus suggests sounds reasonable if you can get Simulink Design Verifier to support those.