tl;dr : no. You do not need anything more than lightroom (in terms of software to process the file)
Transforming from a raw file (that is the pure sensor readings per pixel) to something that you are displaying on a monitor is a rather complex process that involves a lot of math, quite some information additionally to the sensor readings, a pinch of secret sauce, and the turning of a lot of knobs that are up to whoever does the development process.
(note that I am mostly playing with dng files, and some details may be different in vendor specific raw files, but ultimately the underlying principles are the same).
Leaving details such as demosaicing out, the main part of "raw development" is doing the math to transform from the sensors colorspace (like xy), to the target rgb one (like sRGB or AdobeRGB). Here we have generally two approaches:
A matrix (or two). Those transform the whole 3d colorspace into another by shifting, rotating, tilting etc. it. This has some drawbacks: A lot of transformations here are linear, but the sensor responses are not. Additionally the matrix is valid for a specific lighting only (often D55 or D65) and in case of different lighting, sometimes an interpolation (or even extrapolation) of the matrix calculations is done to attribute for different lighting.
A profile table. To account for the nonlinearity of some sensor reponses, there exist calibration profiles. You can find them in lightroom, somewhere in the lower parts of all the knobs. These are mostly large tables that were calculated from special image targets photographed under special light conditions. Also here you sometimes have dual profiles that allow for inter/extrapolation, which contain the data of the same target under different lighting conditions.
The dng format saves these matrices for the raw converter, along with white balance information that helps in applying the matrices. Depending on the camera, this information contains some actual sensor readings, or is just some user setting. In most raw development programs, this will just preset the white balance adjustment sliders! The aforementioned calibration tables are usually part of the raw developer, if present at all.
So how does this process relate to the one your camera creates jpegs with? Most likely not at all. E.g. my canon ixus 950is has a built in table that it uses to calculate the jpeg image, in a way quite different to what adobe lightroom does.
Perception of a good white balance is very subjective and depends on many things, even such as the color of your monitors framing. So you don't like what came out of your tungsten setting? Adjust the knobs until you find the image nice, and resemble what you saw at the point you took it. That is about all what counts. You don't want a graphical representation of sensor values, you want to capture and reproduce a moment, and tell people what you saw at that point. Call it artistic freedom if you want.
But of course starting of with something that is a quite accurate sensor reading won't hurt, so how can be best achieve that? Generally for P&S cameras I found it good to work with the "as shot" white balance setting. In your case, your camera and Lightroom probably just have a rather different meaning of what "Tungsten" actually means.
For best results, you might want to invest a bit in a color checker passport or similar thing to be able to create profiles yourself, tailored to the specific lighting condition you encountered. It can also help finding a proper white point. Just use the included gray card part, and tell lightroom this is neutral color. This can often lead to quick satisfactory results.
And also note that it is called "raw development" (and not conversion) for a reason: you need to give input and tell the process how you would like the photo to look like. Play with the knobs.
