I believe the answer is no. If we could switch to different views in a dashboard, there would be a View > Open (or other view related command)... but these options aren't present once the module is published to the dashboard.
If appropriate, you can permit users to open the module... wherein they'd be able to open the different views.
Another option is to publish all 6 views to separate dashboards
If the views are just filters or hiding/showing different line items, then you could utilize the users list to create a user-defined filter based on a selector (and have the 6 different names appear in that selector). That would really be one view behind the scenes, but may get you close to what you need. What all is different between your different views?
Actually, I was thinking about it a bit more, and I think you do not need a user filter for this. I think it may be a little simpler than that.
Based on what I understand, here's what you need to do:
Create a list with the names of the 6 views that you want to use. I'll call this "Module View Selector"
Create a line item subset based on the module you want to filter. I'll call this "LIS".
Create a module with LIS and Module View Selector as dimensions. Create a line item called "filter" in it, dimensioned as a boolean. Manually check all the boxes that represent the line items that you want to appear in each view (or you can write formulas, if you wanted to get fancy or had views that needed to change by reporting month or something like that).
Back in your original module, you now need to add "Module View Selector" to the "applies to" for the module. You can do not need to add it in the applies to in the other line items (in fact, I recommend that you do not since that will increase sparsity).
Note, if there are user inputs then this solution will not work as it will prevent users from being able to input data.
Filter based on the "filter" line item in the module created in step 3. Publish that to your dashboard. Now, when users choose something different in the "Module View Selector", the view should update and change what line items appear.
Just to add to @bdeaton: I've used LIS to filter based on Org/Dept or User in the past and it only works if the line items are formatted as numbers (because non-number formats cannot be checked). Otherwise this works well. For example, at one client we have an Operating Expense (OPEX) input dashboard, and, depending on which department is chosen, we hide/show input rows so that users only see the rows they need to see. This scales terrifically because most or all departments can use the same input dashboard (but still have a tailored experience).
Also, If the User filter is a subsidiary view, and the data input dimensions are the same as the module (i.e. data inputs are not a subsidiary view), then the filtering will work in a way that continues to permit editing.. Its only if the user input line items are a subsidiary view that they become read-only. User-level filters open up some great flexibility!
One thought, that might work here is to create a separate dashboard for each view. You can then navigate to each based on the selection of a list. It was a feature that was kind of superceded by the contents page, but it's still there
You need to use properties of a list, and when you set up the navigate button, you can specify which dashboard to go to.
Let me know if this is an option and I'll dig out more details