Is there a direct cycle path from Filton to Cribbs Causeway?

I have to collect something from Cribbs mall and I'm refusing to pay £4.80 to FirstBus. All the routes I've checked involved riding on the road? Is there a path without riding on main roads?

I don't want to ride on road because I feel very intimidated by big vehicles. I already hit my head on pavement because of this and ended up in A&E. Still feeling trumatised. Not taking a risk.