@Robert I feel your frustration, never good to be left feeling unsupported. They do appear to attend to problem tickets. Are you getting the feedback with submitted tickets to their support area? If you don't otherwise get a resolution, I'd ping them back via their ticket system and ask for clarification: "can you please let me know where is the incompatibility/issue? What is the name of the 1) template; 2) extension; 3) module; or 4) plugin causing the problem, or is it something else?".
I'm just wondering based on what you wrote, does the problem you see exist if you use the default Joomla site template on your site? If you can, try setting the default Joomla template and see if the problem remains or is resolved. If it's running ok using the default Joomla template then the template you have is probably somehow implicated in the problem (causing it or exposing a problem in Invitex) and you can pursue support with the template provider.
Do you use any of the tools on chrome or firefox to help figure out what might be causing the problem? There are some tools for web browsers that you could use to provide more information to Techjoomla, particularly to capture the difference on your site between when you see those buttons and when you don't; here's a tutorial for
Chrome,
firefox. Of course if you've got a support plan you'd expect to receive a fix or enough information to know where to go next to get your issue resolved. Still it is the case that when you provide as much information as possible to the developer it help immensely with figuring out the source of the problem.
Hope your issue is resolved soon