D. Teodoro de Faria, emmerit bishop of Funchal, published a book on Emperor Karl. https://www.aletheia.pt/products/imperador-carlos-da-austria
In an interview (2013) to Radio Renascenša, the catholic church portuguese radio, he claimed that, several times the IF told him they preferred that the Emperor would remain in Madeira, where "he is revered and honoured, loved and invoked by many christians". https://rr.sapo.pt/informacao_detalhe.aspx?fid=29&did=128298
A visit of the IF to Madeira c. 80 descendants - Adks Karl and Georg, Monika
Not Monika but Michaela
And the party counts with D.Duarte
and Gabriela, Carl-Christian, Michael and many others. https://ppmbraga.blogspot.com/2012/04/membros-da-familia-do-imperador-carlos.html
There is a short video of the IF with the then Mayor of Funchal (today PM of Madeira) http://www.ruigoncalvessilva.com/mensagens-videos/-imperador-carlos-i-da-austria-na-madeira
I am not sure if I am fully correct, but I remember from the time of the funeral of Empress Zita in April 1989 (I was in Vienna for the funeral), talks were going on that Emperor Karl would be brought too to the Kapuzinergruft. The coffin of Empress Zita stood there for a long period on the ground in one of the chapels of the Kapuzinergruft.
Later one, it was assumed Emperor Karl would stay in Madeira, being blessed by then and a reason for pelgrimage.
Is there any plan to return the remains of the last Kaiser "Blessed" Karl to Vienna for burial in the Habsburg crypt? His wife, the Empress Zita, is buried there.