Hello @bsfez (happy new year to you too!).
Access via mobile devices could be facilitated by configuring your router to treat kids.sweethome as a valid domain and pointing it to the IP of your Virtualmin’ server. You would need to use the control panel of your router and add such domain manually every time you need to use one. The manual bit is not ideal.
In order to automate this to a greater extent, why don’t you use a real a domain and subdomains under it to stage the website as kids.yourdomain.tld while it is under development and then use the client’s domain when the website is ready to go live?
See: