My understanding is the Bridging Visa chameleons into and retains all conditions that applied to your most recent previous substantive visa - i.e. in your case your 457.

If this means you won't be able to work as your 457 needs a sponsor, you can apply separately for a BVA with work rights if:

"your current BVA or BVB either does not allow you to work in Australia, or there are work restrictions, but you believe you have a compelling need to work. We might ask you to provide evidence that you are in financial hardship"

Source:
https://www.homeaffairs.gov.au/trav/visa-1/010-

 

My understanding is you could do whatever you want after your current 457 gets expired and then your bridging visa is officially activated. because your 457 is still in place and your bridging visa is inactivated now, your cannot change jobs. but you could do that after August. i think you better ask some qualified agents to confirm this.