I've been using a non-standard bag for ages - through dozens of trips. Not sure how big it is but it's significantly bigger than the normal sandwich type bags. It's about 4cm deep, has an envelope flap fastening with a press stud. It's made of clear strong plastic and I started using it because I was fed up of the other bags ripping after a couple of uses.

I can only think of one time I've been told it wasn't acceptable. I always carry a sandwich bag in the other one just in case.

That said, all my stuff does fit into the sandwich bag so there's no problem switching it if someone objects. Not sure what 'normal containers' you are planning to use but if they wouldn't fit in the sandwich bags you might have some problems.