Hello Ander,
i understand you want to use multiple instances of the same model but should be able to provide different possible parameters for those instances.
If you have R2019a,
There is a new feature called Instance parameters to do this and the doc link can be found here
If you have an earlier version,
In that case As you have rightly described give a Variable name to the Parameter(lets call it with name P on the block name) in the model but Don't define it in the Model WorkSpace of the that model(lets call it model1).
Now create mode2 with model references to model1 with variable P defined in its Model WorkSpace having the value you want.
Similar to the last step create model3 with model references to model1 with variable P defined in its Model WorkSpace having the value you want.
Now create model 4 that has model reference to model2 and model3. This is your final solution.
