Alice "von Wallburg-Habsburg" or whatever she called herself seems to be a fiction. I do not know if she created it herself, or if it was done by her children/grandchildren, but the claim that she was daughter of Heinrich von Wallburg does not jive with the documented life of Heinrich.
Heinrich committed suicide in 1888, age 26. He never married. If he had any illegitimate children, he never recognized them and I have found no mention of them except from the claims of this Alice person (which have all come up very recently). His sister, Clothilde and her husband, Sedul Pegger handled his funeral arrangements and only his 3 siblings were listed as family members.
The cause for the suicide seems to be from a long-standing depression from being abandoned by his father shortly after the death of his mother. The story of the von Wallburg children after the death of their mother, Laura, is not a happy one. They were essentially hidden from view, their father never visited them. The only thing that kept them alive was the good heartedness of their aunt, Baroness von Waideck, morg. wife of Archduke Rainer.
In any event, over several years I have been researching the Wallburg/Skublics family and have now managed to make contact with most of the living decendants (and a couple who have died sonce I started on this). The Wallburg name legally died in 1909 when Franz Joseph decreed the descendants of Archduke Ernst to be illegiitmate and therefore had to use their mother's maiden name of Skublics.
However, Archduke Ernst's son, Ernst (more often called Ernö in Hungary) continued to be called Baron von Wallburg socially, but was legally named Skublics. All records pertaining to his wife and children are in the name Skublics, but his obituary calls him von Wallburg but puts the name in quotes while explaining the legal situation of his name.
The last person descended from the Archduke with the Skublics name, Eva Maria Skublics (by marriage Mrs. Ferenc Pados) died in January of this year, so the name is now extinct from Ernst & Laura's descendants. There are living Skublics around the world, but they are descended from cousins of Laura. The only remaining descendants of Archduke Ernst & Laura von Wallburg are via female lines and all have different names.
No one descended from Archduke Ernst has ever claimed or used the Habsburg name as far as I can tell. When Ernö sued to get his perceived inheritance and name, it was the Wallburg name is was suing for, not Habsburg.
I hope this clears it up somewhat. I am finishing the research on my Skublics book and after one more trip to Hungary to round up a few stray documents and interviews with descendants, I will be writing the book in an effort for a release this coming winter.