"I do not know what can be done about Cardinal Burke. Stripping him of his red hat would only make him a martyr in the eyes of his fans" - well, yes.  But: Pope Francis has the option to fraternally invite Burke to resign the cardinalate, as Pius XI did with Cardinal Billot SJ, who resigned on September 13, 1927 (see http://www2.fiu.edu/~mirandas/bios1911.htm#Billot).

"Better to have given him a desk job with a lot of paperwork like the Congregation for the Causes of Saints!" - for God's and His People's sake, no!!  We would get all the wrong examples of sainthood, ones that are utterly removed from any human experience whatsoever, not at all suited to instill faith in Christ and His Gospel.  Why not make him a one-time only Prefect of the Papal Gardens?  Fresh air outside clears the mind of all canonical worries.

Indeed: May God "change a heart so hardened as the one that beats inside the red robes worn by this canonical conundrum" - Miserere eum et nobis!  Domine, exaudi nos!