Would you mind if I log in to your site and do quick troubleshooting? If this is ok then could you please temporarily create an admin user account and share the account login details privately by checking the option “Set as private reply” when replying to this topic?

Yes, you can achieve this by creating & configuring a new search form and then adding it to the page as described below.

Display Search Form In Page

You can achieve this by adding the below CSS code in the Custom CSS option of Ivory Search plugin settings on the path Admin Area -> Ivory Search -> Settings -> Settings

.is-menu.is-dropdown form.is-search-form {
    left: 0;
    right: auto;
}

.is-menu.is-dropdown .search-close {
    right: auto;
    left: -22px;
}

You can just remove the post type added by the Ivory Search plugin by enabling the search form option as shown in the attached screenshot and then add a new post type by developing a custom code.

You don’t need to use any Ivory Search plugin hook or modify its files.

Then it depends on how you have configured the Default Search Form to search it.

Could you please reset the search form by clicking on the Reset button and then only configure the Search Engine to Inverted Index Search Engine?