Solution: I came up with this uBlock Origin filter rule: startpage.com$removeparam=/abp|lui/, I started by removing all the paramters and then took them out one by one while testing if searches made on startpage.com worked with them: startpage.com$removeparam=/abp|t|lui|sc|cat/. I also got a response from u/SPSupport that the paramters don’t show when I search from the URL bar with my search engine set to https://www.startpage.com/sp/search?query=%s, which I did.
I want to be able to see the search query in the URL, and have a URL that I can always go back to if I restart my browser. I set the HTTP request method in settings to GET instead of POST since that shows the query, but it adds a bunch of extra paramters:
I use the addon Redirector for rewriting urls to youtube etc. to private alternatives. I guess you can use it to rewrite startpage queries.