The above is only the latest attempting at quoting/escaping I've tried. Regardless, I get an error like this when I run the playbook containing the above task:
TASK [Run a script] *****************************
task path: /etc/ansible/playbook.yml:38
<192.168.200.245> ESTABLISH WINRM CONNECTION FOR USER: SOME-HOST\Administrator on PORT 5986 TO 192.168.200.245
<192.168.200.245> EXEC Set-StrictMode -Version Latest
(New-Item -Type Directory -Path $env:temp -Name "ansible-tmp-1466538051.8-95578181041077").FullName | Write-Host -Separator '';
<192.168.200.245> PUT "/etc/ansible/"./files/name\" TO "C:\Users\Administrator\AppData\Local\Temp\ansible-tmp-1466537998.0-243829631467080\name"
fatal: [some-host]: FAILED! => {"failed": true, "msg": "file or module does not exist: \"/etc/ansible/\"./files/name\\\""}