Hi Javed
I will udpate the other post with the missing piece once we finally finish making it work. I did some debugging and I'm writing some notes here, I'm currently still in the debugging process.
The top-left "squares" button still appears, and it does bring you back to the main menu. So you should still have access to the menu functionality, but I understand that it can be less appealing than the left panel.
First, how it works in the "normal" way : the menu is in Mashup PTC.FSU.COREUI.MenuConfiguration.MasterContainer_MU, it gets the menu items from service PTC.FSU.CORE.MenuConfiguration.Utils_TG.GetMenuConfigurationSideMenu_MNC, using parameter SolutionName which in this case is "AMU" (and Enabled is set to true).
The process is similar with the custom navigation method : the menu is inside Mashup PTC.FSU.COREUI.MenuConfiguration.ExternalAccessContainer_MU and it should call the same service. The SolutionName parameter comes from a Mashup parameter, so in the Navigation function (when we click on the button), we would need to add the name, in this case it's "AMU". I would need to add this to my solution in the other post. This should work... except it doesn't and I don't know why yet. For a reason I haven't figured out yet, the SolutionName gets removed, even if we set it in the Navigation :

And because the SolutionName is empty, it doesn't get the menu. Note about this : in the Mashup, the service is called on a Validator condition : The GetSideMenuTrigger Validator says that if there is a solution name then it calls the menu service, otherwise it doesn't.
I'll try to reach a more knowledgeable team to see if they can help me with this. But they are very busy this week so it may take longer.