According to the documentation, exclusion of StripFilter is very easy. It should be enough to set proper value:
com.liferay.portal.servlet.filters.strip.StripFilter=false
The problem is that entry for com.liferay.portal.servlet.filters.strip.StripFilter is missing in my portal.properties - what does it mean, when I cannot find this entry? What is the default value for missing entries?
If your portal.properties doesn't have the entry, it might be an ancient version (?). The current version has this entry and you override it in portal-ext.properties. That file indeed won't have the entry until you explicitly add it.