Mozilla has added a new feature in the latest nightly build of Firefox that will let you choose the search engine you want while browsing in private mode. This way, you will be able to set different search engines for normal mode and private mode.
Download and open the latest version of Firefox Nightly and go to about:config.
In the search box that appears now, paste the commands mentioned below one by one and set the value to true.
Now, navigate to Options -> Search -> Default Search Engine. From the drop-down list that appears below “Choose the default search engine to use in Private Windows”, pick the search engine you would like to use in private mode.
