Hello @Eric Hodges ,
Thanks for reaching out to Microsoft QnA.
Yes, according to my opinion, the second option to deploy application to VMSS using custom script extension looks good to me as it is mentioned in detail in the document.
I am not fully aware of how you should copy the new files to the computer in regards to the third option you have mentioned-doing everything via Azure Automation. I will have to internally check on this.
Please feel free to reach back by replying to my answer and tagging me.
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