I doubt driving the Shark at low voltages is a good idea, which may be why you aren't getting any feedback from anyone who has tried it. No one is wasting their time with such marginal implementations.
The Shark's boost IC is rated at a typical 4A max input, but the minimum max input can be 3A. Wayne conservatively recommends holding the max current draw requirement to 2A. At 2.7V input, you are very close to that 3A limit, and definitely way over 2A. In addition, the Shark is going to run hotter at low Vin and its efficiency will suffer. Thus, an assumption of 80% or 85% driver efficiency may be invalid, and the current draw could be even greater than calculated.
Typically, running at the limits of devices isn't a good idea anyway.
The Blue Shark runs cooler than the Shark and may be a better bet. However, Wayne has stated that his rule of thumb still applies where the full battery range (fresh to depleted) should be greater than 1/2 of Vf of the LED string. Given that, it would seem to me that your envisioned implementation is not a good one. To meet Wayne's rule of thumb, you'd have to run your Panasonic only to about 3.3V, wasting a significant portion of the cell's capacity for a 2A or higher draw.
For a more definitive statement, you probably should post your question to Wayne on the Shoppe's CPF sub forum.