The common practice in current prayer books is to use a meteg to mark the stressed syllable in non-biblical texts if it's not the last one. The earliest edition using this method I have found is R' Heidenheim's siddur (this particular copy is from 1806). In R' Emden's siddur from 1745 there's no such marking yet. On the other hand, in Isaac Satanow's siddur from 1785 all stressed syllables are marked with a meteg, even in case of words with ultimate stress. Do we know who came up with the current system? Alternatively, could you find earlier editions than the Safah Berurah with the current solution?
Heinrich Guggenheim credits Wolf Heidenheim with this system in the introduction to his Haggadah. He writes:
Heidenheim also was the first to print the meteg when the accent is not on the last syllable.
I can't find another place where this is stated explicitly. However, although he doesn't explicitly claim it as his invention, Heidenheim does devote two full pages to explaining his system and the need for it in the introduction to his siddur שפה ברורה (see pages 9 and 10 here).
On the other hand, Yosef Ofer (in an article in Leshonenu vol 64, pp. 297–313) credits this to Seligman Baer in his siddur עבודת ישראל (see the bottom of p. 299). I'm not sure what to make of Baer postdating Heidenheim; perhaps Ofer meant that it was popularised by Baer.*
* He also writes (footnote 8) of a 14th-century Italian mishna manuscript that marks the accent with a meteg mostly for non-ultimately-accented words, ie. the system in question. It sounds like it's not 100% consistent in this respect, so this may not be a proper example. You can see the beginning here, where in the first mishna, תכלת has a meteg in one occurence but not in another, and דרך has no meteg in its first occurence. The third mishna is much more consistent, however.