We have yet to find any reliable construction management app to integrate with QBO UK version. You can explore this first option. You will create an estimate with the engagement letter. Once your client approve your Estimate, the app will generate your invoice automatically.

https://get.practiceignition.com/quickbooks

Another option, utilize this free invoicing app. 

https://go.zoho.com/FuW

You will need this connector to duplicate data to your QBO account.

https://www.integromat.com/?pc=quickbooks