I have been a gold for the past two seasons and plenty of my teammates did not compete back handsprings
Allowed passes without bhs are (that I can think of off the top of my head):
Front tuck
Front pike
Front layout
Front handspring to a salto (or the other way around)
2 front saltos (no twisting saltos)
front salto step out to a roundoff
Front aerial
Side aerial
Roundoff (half turn), Roundoff (This counts, however it is hard to get credit for)
front handspring step out to a roundoff
front handspring step out, front handspring
roundoff to a salto (I know some people who never had back handsprings and went straight to doing back tucks and layouts, that is what most of my teammates who never had back handsprings and are training platinum working on for upgrades)