The different usages of agree are complicated. The Cambridge Dictionary gives four senses/subsenses for the verb, saying that three of them may be either intransitive or transitive. But there is only one transitive example (unless one extends 'transitivity' to include taking a clause as object) provided:
agree verb [ I or T ]
(2) to decide something together:
They agreed not to tell anyone about what had happened.
We couldn't agree on what to buy.
UK We finally agreed a deal.
M-W has 'to settle on by common consent' for this usage. One speaks of 'agreeing terms / a date / a ceasefire / a price ...'.
However, the sense of 'agree' in the question here is 'give mental assent'.
Breaking the sentence down into simpler ones, in either order we have:
Everyone can agree that the evidence we choose shows him being out of touch
We choose this evidence. [With this purpose.]
Note that the 'that' here is the complementiser, not the relative pronoun appearing in the original sentence (substituting 'which' there would make things clearer).
Combining without making deletions:
We choose evidence which is of such a nature that everyone can agree
that it shows him being out of touch with reality.
So we retain the 'agree that' colligation. However, deletion reduces in a single step to
We choose evidence which everyone can agree
shows him being out of touch with reality.
The second variant suggested is ungrammatical.