Ok, this DOES help. As does that mp3, now that I have a chance to listen.
Basically, you DO want to use the bass as a melodic instrument. Couple thoughts.
1) This has to start in the arrangement, and one of the possible valid answers here is (aside from just getting virtuoso good and covering a huge intervalic range in your part) recording both a conventional and a melodic bass part to play at the same time. But, you need to think about, if the bass isn't holding down the low end of your mix, what IS?
2) Honestly, the closest thing to that cello tone I can think of is a distorted bass guitar. This sounds similar enough to the kind of tone you might hear in hard rock or metal. There's a few ways to do this but a very popular one is to basically bi-amp your bassline; either record both a DI and am amped sound, or just record a DI and copy it into a second track. Treat one as a low end track and one as a high end - low-pass your low end track at maybe 250-300hz, and absolutely laughably crush it with a compressor. Leave no semblance of dynamics. High pass your other around 650hz or so, and run it into some sort of fairly pronounced distortion effect - amp sim, maybe, or just a very, very aggressive saturation effect, and if necessary low pass it as well somewhere higher up (probably at least 1.8khz, but potentially higher, especially if you're using an amp sim with a modeled microphone on it, or a real distorted amp you've mic'd up). Then, blend to taste - you get a rock-solid low end, but with a distorted high end combining for a very solid, gritty, punchy bassline.