Your file won't be complete until all required LORs are in, so it won't matter for the schools where the non-science letter is mandatory. Go ahead and submit your secondary ahead of time; your complete date will be determined by when your last mandatory LOR arrives. No sense on waiting to submit the secondary (though you have time to keep editing while you await the arrival of the LOR, I suppose).
Now if the letter is just "strongly recommended" but not mandatory, then yeah, your application could be marked complete and reviewed prior to the last LOR coming in if you met the minimum letter requirements through other LORs.