After searching availability by accommodation type and the search results page is displayed, is there a way to grab the id to use something like mphb_room to grab other information?
If you are referring to the Search Results page specifically, then you may have an additional information displayed there by adding parameters to the [mphb_search_results] shortcode. You may find the list of available parameters under Accommodation ->Shortcodes: https://prnt.sc/10gtuyz.