I think your advice for consulate is right move.

I had my passport stolen on a train a few years back, train was Belgium to Germany and I didn’t report it until Austria. When reporting to the embassy in Vienna they explained that normally background check was required and it would take 2 days, as I had no other ID with me. Fortunately was with a friend who was able to vouch for my ID and as he was also UK citizen they expedited same day a travel document.