Lens for shooting outdoor acrobatics ?

As HCR2016 says, tells us what you want to shoot that you can't shoot with your current lens. For example:

• You might want to shoot subjects further away and/or not crop. There you want a telephoto lens. Since your camera does not have sensor shift stabilization, for telephoto, it is probably better to stick with Panasonic lenses that offer lens stabilization (the longer the focal length of the lens, typically you would need a faster shutter speed to stabilize camera shake without having lens or sensor shift stabilization). Note that with the telephoto lenses in your price range typically need to be switched off with your 14-42mm lens. If you want the convenience of a single lens that covers a wider range without taking off the lenses, there are 3 superzooms (Panasonic 14-140mm, Olympus 14-150mm mark II, and I think Tamron 14-150mm), but these are a little more than your budget. I use the 14-150mm mark II lens often as a vacation lens so that I don't have to switch lenses, but like anything it can be a compromise between convenience and the ultimate image quality.
• You might want to shoot in lower light without using a flash. Typically here, you would want to get a prime lens (fixed focal length). Note if you have never used a prime lens before, spend some time with your current lens fixed to the focal length you want to buy (in this case, probably 17mm, 20mm, 25mm, 30mm, 45mm) and see whether limiting yourself to a single focal length will work for you. If so, go and buy the lens. If not, try the other focal lengths. I have 2 prime lenses, the Panasonic 20mm and the Olympus 45mm. When I use them, I find the 20mm is better for indoor settings in a small room, while the 45mm tends to be better for stage performances, particularly if I can choose where I shoot from. If the prime lens you are interested in is only manual focus, then switch your lens to manual focus as well as a single focal length, and see if you can get the pictures you want. For manual focus, be sure to look at the photos on the computer and not just on the rear screen. Be sure to take your time, as it is a skill you need to hone to learn how far back you need to be for a given focal length, and more so to learn how to do manual focusing.
• Rather than telephoto, you might want more wide angle. There are some options for zoom lenses that go down to 12mm instead of 14mm, which would give you some flexibility that may be in your price range (Panasonic 12-32mm, Olympus 12-50mm). There are also some prime lenses, but most of these in your budget range are manual focus only. If you go for a fisheye lens instead of a rectalinear lens, because sure to understand the kinds of images the lens produces and whether or not your software can 'de-fish' the lens.
• If you want to photograph close-up items, you would want to look at macro lenses. Typically the shorter the focal length of the lens, the closer you have to be to the object. Both Olympus and Panasonic make 30mm macro lenses that are fairly cheap. Typically with macro lenses, you would want to manually focus rather than use auto focus. If your subject is static, you might want to look into post focus mode where the camera takes many pictures with slightly different focus points. You might be able to combine these later via various post processing software (unfortunately the G7 doesn't seem to support focus stacking where the camera will combine the images in the camera).
• I've been experimenting with using the phone app to control my cameras. If you wanted to do it with the G7, you might want to look at either the Panasonic Power Zooms (14-42mm PZ, 45-175mm PZ) or the Olympus Electronic Zooms (14-42mm EZ, 12-50mm EZ) that would allow you to zoom the lens from the phone app. This is obviously a rather specialized usage, and I suspect you aren't looking for it. These 4 lenses are also useful for doing slow zooms in video (though usually it is better to use slow zooms sparingly).
• Since your G7 is not splash proof, it won't matter whether your lenses are splash proof or not (in general -- if you were planning to upgrade, it might matter).

I generally recommend people take notes (either on paper, or with a memo function on your phone) every time you wished you had a different piece of gear.  Then over time, hopefully a pattern will emerge which lens to buy (or buy now with the others on your someday list).