Elizabethan Ayres and Duets Sung in Authentic Elizabethan Pronunciation (Hyperion CDA66003) has 7 Dowland items (plus 2 Campion, 1 Danyel, etc.) sung by Glenda Simpson and Paul Hillier with the Camerata of London, recorded in 1980.

Two very small reservations:

(1) The disc uses E. J. Dobson's reconstructions of Elizabethan pronunciation (which, I think, not all specialists would accept in all details).

(2) The performances, though excellent, sound dutiful rather than spontaneous, and hints of modern vowels still peep through in a few places.

I too have the impression "that the historical accuracy movement does not focus all that much on issues of pronunciation." The same is even more true of Shakespeare: no recording of any of his plays attempts to reproduce the sonorities that he so meticulously constructed. And sonority does make a difference, at least sometimes (consider, e.g., Martinelli's performance of the Celeste Aida recitative as sung by a tenor with a lisp).