Hi Jay,
Thanks for the details, I see the issue.
It is caused, because the menu does not print the search bar upon page load, but it is generated dynamically when the menu ‘magnifier’ button is clicked. Because of that, the search is not properly initialized.
There are two possible ways of resolving this:
1. Try turning on this option on the plugin back-end: https://i.imgur.com/HEjRAZ0.png
2. ..or if that does not work, then this custom code will. Add this custom code to the functions.php in your theme/child theme directory (copy from line 3 only!). Before editing, please make sure to have a full site back-up just in case!
NOTE: I personally recommend solution 2. as it is made exactly for your case. I hope menu creators in the future move away from this bad practice of dynamically putting elements to their menus, as it is causing so many issues.
I hope this helps!