I noticed that blazes are not capable of spawning inside of the smeltery from Tinkers' Construct.
I used a spawner from the nether and brought it back to the overworld.
I modified it using Apotheosis to ignore light level and to reduce the min and max spawn delay.

But the spawner refuses to spawn blazes inside the smeltery.

I confirmed that the spawner can spawn blazes outside of the smeltery without any issue.

I guess that the smeltery blocs (the Seared Bricks) are preventing the mobs from spawning similarly to Bedrock. That was not the case in previous versions of Tinkers' Construct (1.16.5 and previous) nor is it the case when using a standalone instance of Tinkers' Construct (version 1.18.2-3.5.2.40) with an Apotheosis spawner.

Mob spawners need a clear area around them to spawn mobs, so fully incasing your spawner in blocks will not allow spawning, below is my test setup for this and it is working correctly

Also check out vanilla spawn mechanics to understand the vertical spawn areas required
https://minecraft.fandom.com/wiki/Spawner
Even Mitchell's could use some more space open above for optimal spawning

At a minimum you need the level the spawner is on and the level below to get spawns (as shown above)