OT: A stylistic digression (or, let's deal with some *real* nits for
a change)

The sentence in question:

"The USCF has no reason to keep secret material that would exonerate
Truong."

Where I meant:

"The USCF has no reason to keep secret (material that would exonerate
Truong)". (i.e., it's not in the USCF's interest to keep it secret).

But, Phil took it to mean

"The USCF has no reason to keep (secret material) that would exonerate
Truong." (i.e., the USCF might as well throw it away)

How to craft the sentence in question to eliminate the chance of
understanding it the way Phil did?

To me, both "The USCF has no reason material that would exonerate
Truong secret to keep" and "The USCF has no reason to keep material
that would exonerate Truong secret" both sound awkward. The latter
less so, I suppose.