Is there maybe already another way to do this,

Not really what you asked for, but may it can help you: private engines

I reached my goal by manualy changing the following files in /usr/local/searxng/searxng-src/searx/templates/simple/
search.html
simple_search.html

I changed the line

<form id="search" method="{{ method or 'POST' }}" action="{{ url_for('search') }}" role="search">

to

<form id="search" method="{{ method or 'POST' }}" action="{{ url_for('search', **request.args) }}" role="search">

When I now access searXNG using the following url

http://192.168.1.10:8888/search?accesskey=1234abcdxyz!

and hit the search button, the URL parameter is preserved, I can now use this URL parameter to filter access in HAProxy when publishing the internal searXNG backend to internet, and block access if the url parameter is not present.

Is this something the searXNG developers would like to implement or not ?
(I think after a verion update of searXNG my change will be overwritten)

I probably also have to add this url parameter to the preferences, about, stats links (but I could also choose to allow those)

I changed the simple template to preserve existing URL parameters ( preferences, about, stats, 404, links etc). If you are intressested I can upload this modified simple template
The only thing that is not working yet is if I hit the save button under preferences (than it redirects to the root url without preserving the URL parameter)

p.s.

The link "Soure code" in the footer of the main page point nowhere (points to /unknow)

searxng_simple_template.tar.gz
Here the modified simple template for those who are interested (the save button under preferences does not preserve url parameters, also the links in about -> search syntax doe not preserve url parameters)

I think extra/external url parameters should always should be preserved. If someone could fix the save button and also the links in about -> search syntax that would be awsome