Hi there,
I would probably need a bit more context, but let me lay out an approach given my experience with far too many IT Integration projects.
The most common way to look at IT Projects is the project management triangle. So, we'll need to look at reducing quality, reducing scope (i.e. level of integration) OR increasing cost...if our goal is to reduce time.
More likely than not, this client is probably willing to increase cost to achieve these goals. In this area they can do any of the following:
People - Hire more people to get the job done. Get consultants/contractors in, etc.
Processes - Change our way of working. Get more agile. Have better PMO. Etc.
Tools - Figure out which integrations are required, which tools can support our effectors. etc.
Now, in terms of specific ideas and ways to speed up integration, I have no idea until I can ask a few clarifying questions on this case.
Feel free to message and I can take you through this thinking/approach further!