@Mark Goblin Welcome to Microsoft Q&A Forum, Thank you for posting your query here!
You can map a custom domain to a blob service endpoint or a static website endpoint: Map a custom domain to an Azure Blob Storage endpoint
You can use Azure Frontdoor instead of CDN.
It allows a maximum connection timeout of 240 seconds, but if you're write operations are chunkated, it shouldn't be a problem.
Azure Storage Static Website and Azure Front Door
Hope this helps!
Kindly let us know if the above helps or you need further assistance on this issue.
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Please do not forget to "Accept the answer” and “up-vote” wherever the information provided helps you, this can be beneficial to other community members.