Dear Matthew,
Sorry for the confusing and inconvenience.
Project for the web can't be currently connected with the Project desktop client.
However, from the article Project for the web and Project Online desktop client - Microsoft Support, it seems that the product developers are considering on adding more features on Project for the web based on users' feedback.

Project for the web's data is stored in Dataverse, so you may need to use PowerApps to edit it. It may not be possible to connect the data without PowerApps.

Handling data for Project - Project for the web | Microsoft Learn
As our product developers are willing to hear users' requirements, I suggest you submit your feedback and convey your suggestions to our new Feedback platform mentioned in the following article for the product developers to know your suggestions and opinions for product improvements.
Introducing Feedback for Microsoft 365 - Microsoft Community Hub
Please note the platform page will be loaded after some time based on the network environments.
You may access the page as follows to submit the feedback as well.

You may use a personal account or a Microsoft 365 business work or school account to submit the feedback.
Thanks for your understanding.
Have a nice day!
Sincerely
Cliff | Microsoft Community Moderator