By default wordpress does not render HTML within widget titles. They might have some sort of filter or function attached to filter the widget titles, it may only apply to some built-in widgets and/or selected widgets only.
I can think of 3 possible workarounds:
1. Use two widgets, one HTML text widget to display the title, and the ajax search pro widget to display the search bar.
2. Use a text element and the search shortcode within it. I believe their page builder supports shortcodes, so you could use a text/HTML widget with the title and the search shortcode.
3. Use a custom code to render widget title HTML tags, like so: https://premium.wpmudev.org/blog/wordpress-widget-title-html/
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