Problem with displaying field using Views
If this is a Solr/ElasticDB search backend under Search API, you could try this approach.

Create a view mode on the node type (ex. Search Result), then create a template with the html markup when rendered for it. Include all the fields in the display and configure the formatters as you want them.

Next, install Search API View Modes, and configure that view mode to be indexed. It will render that node before indexing it into a new field (search server side). Then, you can create a view display that shows that field, which is just HTML, and bypass some of these issues.

I use this method in every Search API based search I encounter. You're basically storing the rendered HTML on the search server, then printing that as the result.

I can't tell if you are on Drupal 7 or 8. Note that the 8.x branch 'works' but may have some bugs. I have only been able to test simple setups and not use it on production like the 7.x branch.