Here's the thing, though:
While "claim is applied for a military challenge" and "would be killed for military claim" are two different triggering conditions, both are occurring in the same game space in the scenario you describe.
1. When "claim is applied," Seastone Chair is triggered. Once it successfully resolves and activates its replacement effect, you know that when the claim effect for this military challenge gets around to resolving, the chosen character (Tyrion) will die - as the result of resolving the claim for a military challenge.
2. So, once Seastone Chair resolves, you know Tyrion will die when the claim effect gets around to resolving. This is actually no different than knowing that a character chosen as the target for normal claim will die when the claim claim effect gets around to resolving.
3. It follows, then, that once Seastone Chair resolves successfully and the replacement effect is activated, Tyrion currently "would be killed for military claim."
4. Note that the character(s) chosen to die for normal military claim would NOT "be killed for military claim" any longer because the resolution of the claim effect has been changed by the replacement effect. (This is no different than the fact that once successfully saved, a character would no longer "be killed for military claim," which is the reason you cannot continually save the same character in order to create multiple "after a character is saved" triggering conditions.)
Taken all together, this means that once Seastone Chair successfully resolves, creating the replacement effect, Tyrion would be killed for (replaced) military claim - satisfying Pod's triggering condition.
Don't mistake the fact that the "claim is applied" triggering condition still exists once the replacement effect is applied as indicating that the original claim targets would still "be killed for military claim." Once the replacement effect is successfully applied, the original claim target(s) is no longer in danger of being killed - which is no different than how saving a character once shuts down all need (and opportunity) to save it again.