Is there a way to specify how "types" are ordered?

It's not directly supported, but I think you can get it done using an additional field, with values 1_preprint, 2_misc, 3_..., grouping by that, and then using CSS to hide the group headers and show custom headers instead:

div.bibbase_group > span:nth-child(2) {
  display: none;
}

#group_1_preprint .bibbase_group_count:before {
      content: 'Pre-prints';
}
#group_2_misc .bibbase_group_count:before {
   content: 'Other Publications';
}

I know it's not ideal, but it should work.