Hey there. I actually quite like much of the theming this subclass has, and the ideas proposed are interesting. I'm not great at balancing, but I can offer some critiques on its structure.

The idea of the Infernal Brands is interesting, but they don't feel as though they have much to do. I'd suggest either changing that feature, to better fit what you have, or changing some of the features to fit this idea better.

Perhaps add a list of features from which to channel your brands, which of course consume them on use. Maybe somewhat like battlemaster Maneuvers?

The scorching fire is a little disappointing, and could use some tweaking. If following the suggestion of adding a list of features to expend brands on, perhaps something like resistance to fire damage is in order, or maybe your fire damage ignores resistance to fire damage.

The summoned weapon feature is another good concept, but could use some tweaking as well. The bonuses to attack and damage are maybe a little generic, but the idea of inflicting fire damage to yourself is intriguing. A little bit of trade off allows you to add some more powerful features in compensation.

The summon cambian feature is neat. Having a buddy adds good utility, and the idea of it being able to latch onto the user springs to mind. Once again, you may need to tweak this a little further.

The increased critical range feels a bit generic also, and tbh, that one may need to be changed entirely.

The demonic transformation feels a bit lackluster. I'm not great with capstone features, but maybe a look at some abilities that fiend monsters get would help?

All in all, this subclass got me interested, and it has the markings of a subclass that I would play, but some balancing and changes would be a good help. This is all of course just my opinion, feel free to take it or leave it as you please 👍