I don't think you need to spend more money. The moon is pretty easy to shoot, although a 600mm lens does require care in setting up. The tripod and head you have should work for lunar shots.
Some tips:

• Mind your shutter speed. My starting point for moon pictures is using manual mode, ISO 200, 1/160 shutter speed, f/8.0 aperture. Adjust parameters as necessary. There is no need to go below 1/100 shutter speed unless you're shooting a lunar eclipse series. Learn to use your histogram and zebra stripe mode (flashing highlights) to set correct exposure.
• Keep your tripod legs as short as possible. This minimizes the tripod from doing the hula (as in the Hawaiian dance). I sit on the ground when doing astro shots from a tripod so I can keep the legs as short as possible.
• Set your tripod on stable ground (e.g. concrete, hard dirt). Grass is not stable.
• Keep out of the wind as that will cause movement of your setup. You can use a wall, car, bush, ... for a wind block.
• Use a remote cable release. These are relatively inexpensive, especially if you get a third party product.
• If you don't have a remote release cable, then use the exposure delay mode (Nikon term) or the self timer mode. The camera takes a picture a few seconds after the shutter button is depressed. This allows the camera shake to die down after you depress the shutter release, and for DSLRs, the exposure delay mode also reduces mirror slap vibration.
• Mount the lens collar foot to the tripod head. Do not mount the camera body to the tripod head as it is off balance and will stress the lens mount when using a large & heavy lens.
• If vibration is still an issue, enable OSS. Generally, OSS shouldn't be used on a stable setup as there is a small amount of noise from the servo system that stabilizes the image.