one possibility is the use of 'size exclusion' by 'high pressure liquid chromatografie' (HPLC) but on the one hand it is pretty tricky to get reliable results and on the other hand, who has a HPLC at hand.
Another possibility is to use the change of the temperature either of the freezing point or the boiling point. The 'Ebullioskopie' (german word, possibly something alike in english) is the method to evaluate the average molecular weigth by change of the boiling point.
A short description with formulars in LATEX code:
In analogy to Clausius-Clapeyrons equation for the vapor pressure curve of a pure liquid
\begin{equation}
\frac{dP}{dT} = \frac{1}{T}\frac{h''-h'}{v''-v'}
\end{equation}
there are equations describing the phase boundary for binary mixtures, but also with constant pressure there is a difference between the boiling temperature and the dew point temperature. Changing the concentration of the second component results in a change of the boiling temperatur following the differential equation:
Adding a small amount of unknown molecular mass to a solvent with much smaller vapor pressure, it is possible to quantify the molar fraction by messuring the rise of the boiling temperature. From the molar fraction you may calculate the mean molar mass of your unknown substance.
To measure the change of temperature you either need a very precise electronic thermometer or a 'Beckmann Thermometer' that may show the change of temperature only but with 1/100 °C precision. Further you need some glas equipment, where boiling should take place some way distant from the reflow of the condensate.
Some hint: The oil should be water free and temperature changes are quite small, but anyway keep the amont of oil small, e.g. below 1/1000
Much luck
Jörg Ho
PS: You can expect the freezing temperature method to give greater temperature differences, but I never worked with it.
Determination of MW of oils is based on the assumption that the given oil content is made of stable TG or fatty acid. Fish and chicken oils are mainly unsaturated fatty acids. So, Waste frying oil from these sources is a mixture of free short chain acids, hydroperoxides, volatile compounds, some saturated fatty acids... So the assumption is not longer respected.