Hi @Mike_647
Can your employees set GPO on their computers? If they can, you can ask them to set IE mode policies according to this doc. After setting the policies, the site will always load in IE mode without other operations. But the problem is that the employees are not in the same domain, so you can't use a Domain Admins account to distribute the policies. The employees need to set the Local Group Policy on their own computers.
If the setting group policy way is out of consideration, you can only use the above two ways you mentioned in the question.
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Regards,
Yu Zhou