Wanted to follow up on my previous assumptions, while you could build a beehive kiln out of fire clay, you also need the kiln door, which requires four nails and straps, and two iron bars, which gates it firmly behind the Iron Age, when this really is something you should be able to make around the late Copper/early Bronze ages.
Really, the amount of resources required to put one of these together is ridiculous. You need 250 bricks of your choice of tier (12 for the door, 18 for the gratings and 220 for the brick blocks), and at least 17 ingots of iron (1 for nails and strips, 4 for the bars for the kiln door, and 12 for the iron hatch doors). Unless you want to double the amount of fire clay required, you'll also need two ingots for iron pounder caps for your pulverizer in order to make higher tier bricks. You'll also need a full windmill to power that pulverizer reliably, which is 256 Flax fiber. You can get it to work on a particularly windy day with half that number, but 128 flax fiber is still difficult to come by without getting good luck with cracked vessels. All this just to not use up a little grass, a few sticks, and some firewood is absolutely not worth the amount of effort that goes into it. Especially since the bricks can, and will, break.
I think the Beehive Kiln needs to be reworked to only use fire clay bricks, and not need iron to make it. You can easily replace the iron bars with solid stone, like that used in the Quern and Coffin recipes, and the iron nails and strips with mortar. Then, reduce the heat damage drastically, and only let firewood be usable as a fuel. This makes it accessible for the mid-game, which is where it will be most useful. That's when you'll want to fire a bunch of molds for things like hammers, anvils, helve hammers, prospecting picks, and general tools and armor. Iron is still far off, even if you're in the early bronze age, but you can reliably make any ceramics you need without needing to aerate your lawn.