You are running into 2 separate problems. And you are on the verge of solving both.

Dynamic Range

You already are playing with the exposure settings. Which basically will solve this issue. Your eyes have an enormous amount of dynamic range in light which means that your brains copes with a large difference between the light and dark portions of an image. However, your camera is a bit more limited.

As you already found out, any color will get white if it is bright enough. This is what you are experiencing here. The yellow light of the lamp is so much brighter than the background that it will appear whitish at the lamp's hot spots.

To solve this there are basically 2 routes:

You can try to bring the bright and dark areas closer together by brightening up the background with an ambient light. You tried doing this with a flash. This however can introduce problems in the white balance. See part 2. (If the lamp is dimmable, you could also make it less bright)

You can also try to make several exposures and combine them im Photoshop afterwards, basically generating an HDR (high dynamic range) image. So you would expose one shot for the dark areas, one for the mid-tones including the darker areas of the lamp and one in which the highlights of the lamp have the desired exposure. Then combine them.

Color balance

Another problem that you will encounter is that your eyes are incredibly well at doing an auto white balance. Your eyes will see a lot of differences in warm and cold light as whitish.

So if you use a flash to narrow the dynamic range of the scene, you will get mixed light. You have a daylight balanced flash and the very warm lamp light. This will look off in the final image.

Your idea to use a yellow filter is absolutely the right solution. These filters are often referred to as CTO (Color Temperature Orange) and will shift the flash towards a more orange shade, which will fit better to the lamp.

Your camera however does not know any of this. It will try to find anything white and then skew the colors to that. You should set the color temperature in the camera until you are happy with the tone of your lamp's light. Then add as much CTO (the filters range from steps like 1/4 to 1/1 CTO, sometimes also called quarter, half or full CTO) to your flash that the flash fits to the lamp.

Then you will get the desired image.