Manually removing database records of a module
I second the voices above.

My recent experience to get out of the pdoexception is crude and more akin to a 'big hammer' rather than anything elegant. 


make a backup of your database and the code   
either use a UI like phpmyadmin OR sql if you feel more comfortable with code
create the table it says is missing (website_staging.search_api_server)
add at least 1 variable (id)... take the defaults of integer
you don't have to add a value to the table, just create it and add the definition for the id variable
try your uninstall again


In my case for my custom module practice development, I also had to add a second variable because there were two 'keys' on the table.  But I'd try this with just the one variable (id)... hopefully it will be enough to get you out of the PDOexception and onward to module/code review.

The concept here is to give Drupal the minimum info it needs via manual PHP/SQL creation so it can 'connect the dots' and move forward to execute your commands (uninstall).
JW