The first step for the integration I think you want is to greatly simplify your culture and project plans.
The metaphors to simplify your culture include "agreement flywheel," "expectation factory," and "decision symphony."
Many project plans contain "Infinite Activity." Instead, build a project plan that contains strictly your "Finite Productivity."
Teams have infinite decisions to make, but when teams "bundle" their decisions, teams can land on a finite number of "Decision Bundles." My default is 54 bundles. Separately, I can share what these are, but experienced project managers are already familiar with many of these (under the umbrellas of people, process, and technology).
Organize your Decision Bundles into a schedule / project plan that reflect interdependencies. For every Decision Bundle, create four rows. 1) Draft. 2) Review & Revise. 3) Approve. 4) Distribute. Assign names to each row (just like you would a RACI matrix). Add duration estimates. Default to 1 day for each row. If someone asks to reserve more than 1 day for their assignment, no problem.
I'll jump to what happens in the Lessons Learned. You only have 5 places to reflect Lessons Learned.
Decision Bundle count and names (e.g. increasing or decreasing from 54)
Decision Bundle contents / insides
Interdependencies of the bundle (sequential / parallel)
Assignments among Five Verbs (Draft, Review, Revise, Approve, Distribute)
Duration
That's it. Don't manage anything else because it's inconsequential work not worth managing. Fair? I welcome reactions, questions, ideas, etc.
Back to the concept of Finite Producitivity ... 54 x 4 = 216. That is my default "Finite Productivity." A mature team knows its metric, and knows that every Lessons Learned stands to fine-tune this metric.