Hello,
Sorry to say but we have confirmed from the Test Site, the Mega Menu through HTML Block selection is working fine so this clears the fact that it is not a Theme end issue.
On your Site, the Selection of Fullwidth or Set Sizes for the Parent Menu Item is getting automatically changed to default after Saving the Menu. So, this seems to be a third party Plugin conflict on your Site. Can you please once try deactivating all the third party Plugins on your Site. If the issue gets resolved, then reactivate them one by one to see which Plugin is causing the issue.
If the issue still persists, then please take complete Backup of your Site first, then provide us the permission to completely debug your Site which includes deactivating all the third party Plugins and removal of on-Site Customization changes too, so that we can thoroughly check this issue on your Site and assist you accordingly.
Best Regards