I would love to know if we have a single word that can be used in general speaking and writing in place of "But anyway".
Yet is a somewhat rare alternative to but, yet it is well suited to the usage in question:
yet – adverb 3 – Nevertheless; in spite of that:
‘every week she gets worse, and yet it could go on for years’
And may be omitted before yet, as in the first sentence of this answer, leaving an equally acceptable variation of the dictionary example:
‘every week she gets worse, yet it could go on for years’