Without empty cores, you're pretty much done before you start.

The only real option I see is to take the bulk loader into the darkroom and hand roll a quantity of film that will fit into the loader's supply chamber and thread it in. It'll still work in the loader (I think), though you may find the remjet gets scratched up as it rides on the film's back surface inside the loader chamber.

If you know anyone else who bulk loads, you may be able to bum some empty bulk spools to roll onto.

On potential option that just occurred to me is a method I've heard others talk about, but haven't tried myself. Since a 36 exposure roll of 35 mm is about 150 cm long, you could just (in the dark) pull the end off the big roll with one hand, run the other hand down the film to your full arm span, and hold the film there to be cut. Then hand roll the strip into a cassette (and don't forget to capture the new end of the big bulk roll and protect it before switching the lights back on).

There is a small risk here of getting a strip that's too long to fit in a developing reel, but you could use a reference stick or notches on the edge of the darkroom counter for a length reference to prevent this.