How can the mass of the sun be determined without using Kepler's law or gravitational lensing? Can the mass of the sun be determined using the abundance of elements or alternative methods? It would be better to ask the question like this if we assume that I have an object that contains baryonic and non-baryonic parts (such as dark matter), how can the difference between the two be obtained?
 A: You can make an estimate of the Sun's mass using helioseismology - i.e. examining it's radial oscillation frequencies.
Broadly speaking, things oscillate on timescales that are, at least dimensionally, given by $(G\rho)^{-1/2}$, where $\rho$ is the density. Providing then that you have the radius of a star, the pulsational oscillation spectrum gives you a route to making a mass estimate, though the details will still be to some extent model dependent (e.g. Lundkvist et al. 2014).
Helioseismology has been suggested as a means of probing for dark matter in the Sun (e.g. Cumberbatch 2010 ), though not by estimating the mass, since there is no effect - one source of gravitational mass is much like another. The structural effects are likely to involve increased energy transport out of the core, requiring a lower temperature and higher density. This will change the sound speed profile and acoustic oscillations and reduce the neutrino flux.
A slightly less accurate way is to simply use a stellar model, combined with an age from radioisotope dating of meteorites, to estimate the mass of the Sun given its current luminosity, radius and composition. In practice this would depend on model fudge-factors like the convective mixing length and overshoot parameters, which are normally fixed by assuming the Sun is an exemplar for a star of its mass!
A: You can make a model of the Sun. Its spectrum, radius and composition are known from various astronomical measurements. I am sure the mass will be a sensitive parameter. Any dark matter present will not be included in this result, M$_1$. Then you can compare the result to what you find, M$_2$, from the dynamics of the solar system. Any significant excess - accuracy is a concern - of M$_2$ over M$_1$, would indicate the presence of dark matter. This is probably what you have in mind. It can be done and possibly has been.
A: It always comes down to Newton:
$F_{cpt} = \frac{mv^2}{R} = F_G = \frac{GmM}{R^2} => M = \frac{v^2R}{G} = \frac{(2.978\cdot10^4m/s)^2\cdot1.496\cdot10^{11}m}{6.67\cdot10^{-11}\frac{Nm^2}{kg^2}} \approx 1.99\cdot10^{30}kg$
A: This is an interesting question, as it's always the product $GM_s$ that's measured, by Newtonian theory or even gravitational lensing.
It would be good to know the mass of the sun, if possible, as it would enable the gravitational constant $G$ to be determined more accurately.  At the moment it's the physical constant that's known least accurately, to 4 significant figures.
https://en.wikipedia.org/wiki/Gravitational_constant
The mass of the sun will be known to the same level of accuracy as $G$.
The abundance of elements depends on the age of the sun as well as mass.  The luminosity of the sun also depends on mass but as far as I know cannot give a more accurate measurement than the gravity methods.
These methods use computer modelling with known planetary orbits and the mass of the earth.  The mass of the earth needs measurements of $G$ and satellite or Lunar Laser Ranging (SLR or LLR) - and $G$ is determined by updated versions of Cavendish's torsion pendulum experiment
https://en.wikipedia.org/wiki/Cavendish_experiment
P.S. after the question edit: the gravity methods would give quite an accurate value for the combined amount of matter (normal plus dark) that is in the sun.
If there is a way to determine the mass of normal matter, by luminosity of the sun for example, or as you say abundances of elements, then the amount of dark matter could be deduced.  The problem would be that the luminosity-mass relation would be calibrated by observations of other stars that may contain a similar proportion of dark matter - perhaps computer modelling using normal matter in the program would be the way to go...
