Your question mentions a small brass brazier, but Find Familiar doesn’t call for a small one. The spell calls for a brass brazier which can hold 10gp worth of charcoal, herbs, and incense. It doesn’t specify the volume of burnables or size of the brazier — however, if it wasn’t very much, the spell would have said “a small brass brazier” too, like you did, to indicate that it’s portable. It doesn’t though, strongly suggesting that it’s a normal brazier of the inconvenient sort suitable for a magic ritual, not an especially adventure-portable one.
So, what’s a normal brazier, large enough to contain the spell’s materials? This is:
Scale model of medieval brazier, by Linda Sweigart of CalicoJewels
It’s about the size of an end table. It’s the sort of thing a warlock might stand over while casting magic rituals.
The thing is, a brazier in a fantasy context is patterned on the braziers in our own history from back when homes and castles were heated with wood or coal, but multi-roomed buildings didn’t have fireplaces in every room. It’s basically a portable room heater — but only portable in the sense that a normal person could pick up a small table and move it to another room. They’re furniture, not adventure-portable.
So if you’re looking for a normal brazier to make emergency castings of Find Familiar convenient, you’re looking at the inconvenience of lugging around bags of coal and a piece of medieval furniture.
All is not lost
But all is not lost! This is fantasy, after all, where wizards fly airships held aloft by magic and shruganium, and gnomes build clockwork contraptions with no apparent fuel source that can knock down a house. Or at least, some games of D&D are that kind of fantasy.
If your DM is running a more fantastic game of D&D, ask them about finding someone who can build you a special, collapsible brazier. Maybe with a bowl made of rotating segments that cleverly fan out and collapse into a wedge in a circular telescoping action, and legs that can fold up and be strapped to your backpack. Who knows what that sort of thing would cost or how long it would take to craft — ask your DM about these kinds of interesting details.
Or maybe your DM will look at this request, taken a bit aback, and literally wave their hand and say “don’t worry about it, just write down ‘brazier’ on your sheet and let’s get on with attacking this giants’ castle, okay?” Not all DMs care about these details, or accurately handling medieval room furniture.
We don’t know your DM, so the critical information about whether this is important to them, in their world, is unavailable to us. I can only tell you superficial things, like what a brazier is, and that D&D 5e doesn’t have any guidelines for how long they take a blacksmith to make or how much they’ll charge (leaving even that detail back in your DM’s lap). So this comes down to:
Ask your DM. They’re the one you have to play with.