Short answer: NO. 'That' as the relative pronoun cannot make noun clauses and stands as the subject of them.
Long answer: It seems you've confused 'that' as the relative pronoun with demonstrative pronoun 'that'. The underlying structure of your sentence is 'I know that that is a soldier.' You could replace the second 'that' with 'he' without changing the meaning of the sentence.
'I know that he is a soldier'
The first 'that' is NOT even a relative pronoun. It's called a 'complementizer.' However some consider it a relative pronoun but in either case it's not the subject of the clause.
https://en.m.wikipedia.org/wiki/Complementizer
I assume you already know the difference between an 'adjective clause' and a 'noun clause.' 'That' can function as the subject of an 'adjectice clause' as an instance,
'I know a soldier that is tall'
'that' here cannot be replaced by 'he' since it would be grammatically wrong.
*'I know a soldier he is tall'
Let me give you some more examples of noun clauses with different relative pronouns.
You can say 'I know who did it' and here 'who' is the subject of the relative clause but you can't say '*I know THAT did it.' Other examples would be
'I belived what happened there was a tragedy.' NOT 'I believe THAT happened there was a tradegy.'
'I learned which/what can be used safely in this recipe.' NOT 'I learned THAT can be used in this recipe.'
(Here the sentence would be right if we thought of 'that' as a demostrative pronoun. Obviously if we do this the meaning of the sentence will be changed. It would mean: I learned IT can be used safely in this recipe)
'Who is to blame doesn't matter now.' NOT 'That is to blame doesn't matter now.'
I know what made the noise last night. NOT I know that made the noise last night.
The whole noun clause can work as the object or the subject, nothing new about that, but the relative pronoun 'that' can't be the subject of the noun clause.
As I mentioned above, I can think of clauses in which 'that' is the subject as in,
but it's not a noun clause but an adjective clause.
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In a nutshell, in noun clauses with 'that' you have to follow this order,
....that + subject of the clause + verb
In other words 'That' cannot be the subject of a Noun Clause.