The method you use will depend on the accuracy you want. As you've used the word 'estimate' in your question title, I presume that you don't need standards lab accuracy. Adding an accuracy specification, or adding the purpose of the equipment to your question, may get you more relevant and focused answers.
The simplest way to estimate is to measure the resistance when cold, and at temperature. This depends on having a graph of resistance versus temperature.
This data is for pure rhenium from Kaye and Laby at the npl.co.uk via the wayback machine
temp K resistivity (10^-8 ohm.m)
There will be errors from the purity of the element, and the fact that the measured resistance is an average of the cool bits of element near the mounts, and the hot bits in the middle. There may be a steady drift in the diameter and so resistance of the element due to degradation of the element, though this should be recoverable by a new cold measurement. There may be a drift in the resistivity of the element material, due to reaction with the atmosphere it is operated in.
The main alternatives are infrared thermometry, and spectroscopy. The latter may be possible fairly cheaply with a diffraction grating and a camera.
For the roughest and dirtiest way, look at the colour of the filament, and interpolate from colour description versus temperature. You will find many tables of colour versus temperature online, generally from artisans who weld, operate kilns, heat-treat metals. Spectroscopy is just a way to instrument this process and make it objective rather than subjective.