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I think probably making the bracelet long enough to loop through the buckle and then stitching it together with very strong thread would be your best bet. That is how they join a regular collar around the buckle ends.

Oh,it's amazing,I didn't know about lark head knots *__*
Ok,I'll do it! And for the other part,I guess she used some glue,maybe?

i made my cat a collar and i made sure you can slip at least two fingers between the cat's neck and the bracelet. I closed the bracelet with a "button" clasp.

i really want to try but i dont know how to do it

Using lark's head,but there is even another way.Tying some more,try to make a seamless effect (there is a tutorial that explains how).Unless your pets will pull on it REALLY hard it won't break.It's simple,keep some threads at beginning,and then put them around the beginning of the end of the endings,then when you made a spin of it,you knot it so that they will all get tied together.Sewing or glue will make it even stronger.Hope i was helpful

Maybe try getting an old pet collar, cut the ends off it and sew both ends to your braiding work. Make sure the collar is the right size for your cat!! It won't be very good if he can slip out of it!! :)