I think he should have written and told us. Honesty is written on his face.

Past Participle: Past Participle used to indicate the actions that took place before some point of time in the past. Past participle uses the V3 form of the verb. It does not stand alone and uses an auxiliary before it.

I think he should have written and told us.

Here, the sentence is using 'should have' shows a wish that did not happen. The V3 form of the verb is used with such modals, so 'written' is used.

Honesty is written on his face.

Since 'is' is written before the blank, which means it is in the passive form, so the V3 form of the verb 'write' is used, i.e., 'written'.