After some tinkering and simulation in LTSpice I ended up with the version of composite mod below. This is similar to the most common found online. This works fairly well on my CRT TV. It has a bit of noise but it is playable.
SIDE NOTE: I later realized that some of the electrolytic capacitors in the Atari 2600 had to be replaced. If you do this mod make sure to fix this also!
I also wanted to be able to connected it to my RetroTINK 2X Pro HDMI
converter. I had problems with getting any image at all and
when it worked the image was very dark.
I measured the amplitude from the signal and it was about 1/4 of the ideal level with a 75 ohm load. This is expected since this composite mod is an emitter follower and provides no amplification of the signal. The purpose is instead to be able to drive the 75 ohm impedance.
I came to the conclusion that the signal needs to be amplified. I took inspiration of some designs that I found online and made a spice model of my idea. After some iterations of testing and simulation I ended up with this design.
The only issues is that the colors are slightly wrong when capturing the HDMI signal. But that can be adjusted with the capture software. The amplifier also removes a lot of the noise on the CRT. For the interested... here are some LTSpice simulation results that I saved.