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If you are VERY careful with their placement then french cleats would work well at supporting that with the added bonus (assuming you only want to have a shelf for support) that you could dispense with the supporting shelf. One fixed to the bottom rail of the frame to support the base and one fixed to each upright about 2/3 of the way up.
Alternatively these floating shelf brackets are good. Easy to install and strong.
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Looking at those pictures, you could happily add a backing board to add a bit of strength.
I'd advise against a split batten/french cleat as mounting it will be a pain in the arse/nerve wracking in terms of avoiding the mirror with the screws.
I'd go for Rymans or strap hangers into the frame at the edge, and sit the frame either on L brackets at the bottom or the wood shelf you mentioned before.Link for Rymans:
https://absoluteproduct.com/ryman-hanger.htmlLet me know if you go for Rymans and I can hopefully offer some pointers - they can be a bit of a devil to use, but there's a couple of tricks to make life easier.
have it rest on a very solidly built shelf (that's the weight taken care of)
then to stop it tipping forwards, there are tonnes of options. how chunky is the frame (not talking about the board at the back? my best idea is to put hooks in the frame and use picture wire to take some of the weight. it's adjustable, so you should be kosher.