I was thinking about that this past Friday Night, as we sang Lekha Dodi to the tune of Eli Tziyon at shul, and I realized that it's not quite that simple. Lekha Dodi interweaves two themes — one is the approach of Shabbat, and the other is aspirations for Redemption. Matisyahu got the title of his album “Shake Off the Dust... Arise” from Lekha Dodi, after all, where it refers to the Jerusalem and the Jewish People, identified with each other, shaking off the necrotizing chains of exilic lethargy and rising to meet the Messianic Age.
Lekha Dodi is fundamentally hopeful — just as we stand at the end of the mundane week, on the threshold of the transcendent alternate universe of Shabbos, it looks out from Galut forwards, hopefully towards the Coming Days, and leaving the Valley of Tears behind.
When we foreshadow Tish‘a B’av by singing Lekha Dodi to the tune of Eli Tziyon, we aren't taking the mournful nature of the Fast and applying it to Shabbat where it doesn't belong — we're subverting and undercutting the entire mourning process before it's even had a chance to really get started. We go into Tishabav confident and hopeful, looking forwards towards the as-yet-unfulfilled prophecies of Consolation, instead of just looking back at the already-fulfilled prophecies of Calamity. Long before we read Eikha, we've already spun its message.
Complete and utter side-point:
You can also use the tune of Eli Tziyon to sing the song of the Dwarves longing for their dispossessed homeland in Prof. JRR Tolkien's The Hobbit, both the English original and הנעמי's Hebrew translation. Which is pretty cool, considering the similar themes.