"Lay bare" is actually a set phrase (https://en.wikipedia.org/wiki/Set_phrase). You can see how it is always listed separately in dictionaries, for example here:
Although it can still be used in its literal meaning, as a combination of two separate words that together mean "to lay something out after stripping it of its covering", this is becoming somewhat archaic. Here is an example of this literal usage:
But Peace destroy'd what War could never blight,
And laid those proud roofs bare to Summer's rain
(from Childe Harold's Pilgrimage by Lord Byron)
To a modern English speaker, this will most likely sound too formal or old-fashioned. Nowadays it is usually used only in the following way:
to revealed or uncover private information or feelings
The two example sentences you used are both quoted in the Merriam-Webster definition above.
So in modern use, the phrase is not really treated as an arbitrarily constructed [verb] + [predicate adjective] combination, but rather as a single lexical item. As such, it is resistant to being broken up or rearranged. For example, the combinations: "lay/laid bare the truth/secret" produces hits consistently in google ngrams, while the combination "lay/laid the truth/secret bare" produces no hits at all for the entire modern era (1900 to 2008).
(I suspect that the reason "lay bare his soul" and "lay his soul bare" are both relatively common is because of the very similar idiomatic phrase "bare his soul". There might be some cross-contamination between the two phrases, and since "bare [pronoun] soul" is a set [verb] + [posessive pronoun] + "soul" combination, the construction [verb] + [posessive pronoun] + "soul bare" ends up sounding ok by association.)
In general, the rule you brought up would apply to a more loosely-coupled construction. For example, the phrase "make [something] public":
He made the information public. (OK)
He made public the information. (awkward)
The book is an attempt to make the secrets of this very powerful
political family public. (ok, but could be rearranged to assist readers in parsing the sentence)
The book is an attempt to make public the secrets of this very
powerful political family. (ok--the awkward construction becomes acceptable because of the length of the predicand noun phrase)
However, when it comes to "lay bare", because it's more tightly coupled, the syntax is more similar to that of phrasal verbs, such as "lay down" or "lay out", which normally stay together and only allow the insertion of relatively light noun phrases. More specifically:
Particle phrasal verbs that are transitive allow some variability in
word order, depending on the relative weight of the constituents
involved. (see examples here: https://en.wikipedia.org/wiki/Phrasal_verb#Shifting)
Here are some more verbs as part of set phrases:
taken aback (by): no longer interpreted as a freely constructed [verb] + [adverb] combination. Thus you might still say "taken completely aback" (although even this is questionable) but never something long like "taken completely, and yet unsurprisingly, aback".
set foot (in/on): no longer interpreted as a freely constructed [verb] + [noun] combination. Thus you would never say something like "I have never set my foot in her house" or "they would never set their feet in her house".