On my bikes, I file down the heads of the original rivets with careful and judicious use of a rotary tool cutting disk, until I can tap the rivet spine out the back of the head tube. I replace them with faux rivets-- essentially the heads of small brass nails, glued in place:
This is a Raleigh headbadge that I used this technique on, to give you some idea of how this can turn out on your Raleigh mixte:
I've applied the same technique to other headbadges, as well:
I always glue the headbadges back on, and glue the faux rivets separately.