I have just tested a Toolset only solution using a Single Page Content Template, rather than a Beaver Themer Singular Template, and the edit form worked as desired. When I choose to "edit this post," the edit form overtook the page. Only the form was visible and not the post itself.
Are you saying that if I use Beaver Themer Single Template, I will only be able to show the edit form within a Post Content Module?
Regarding the conditional logic used in my Beaver Themer Icon Modules, this is new feature only available in the Beaver Themer alpha release 1.2 which can be downloaded from the Beaver Builder website. hidden link
I took a look at this and I surmise that the issue arises because Beaver uses a hook that is triggered on the back-end but not on the front-end when saving posts.
They are not allowing for the possibility that users may save posts from the front end: Toolset Forms is just one of several plugins that enable front-end post editing, and I expect you would see the same if you used an alternate forms plugin for front-end post submissions.
That is useful feedback you can give to Beaver.
In any case, as this is an alpha feature, meaning it may be some months before it even makes it into a beta and is subject to change, our developers wouldn't look at the issue at the moment.