As a Nikon shooter with the D5300 that used to use the Nikkor 300mm f3.5-5.6, I don't think you should buy an x2 teleconverter.
The D5300 already has really bad autofocus. Adding a teleconverter into the mix will only make things worse. Teleconverters can give you more reach but it also reduces the amount of light that reaches your camera, which makes it harder for the autofocus to do its work. Also, most teleconverters are made with cheaper glass, which can make your images less sharp and fringy. The Tamron 70-300mm is a full-frame lens, adding a x2 TC would make it more than 900mm. This might seem great, but in reality, because of how long the focal length is, it means that even the smallest vibration will be magnified. To get any sharp images, you'll have to raise your shutter speed by a lot. When the aperture is so small already, having to lower your shutter speed will make your images even noisier (and the D5300 just can't handle low light without making it unbearably grainy).
It may be a smarter option to just get a new lens. For example, you can get the Nikkor 200-500mm f/5.6 used for around $700 (in Japan, at least). In my opinion, it is the best budget lens for wildlife photography - it's relatively very cheap and still performs exceptionally. It's a full-frame lens, so it is around 750mm and has 4.5 stops of exceptional IS. The combination of having f/5.6 and 4.5 stops of IS allows you to use lower shutter speeds and ISOs (and therefore less noise). I own both the D5300 and the 200-500mm myself, and it works perfectly well with the D5300 (although the camera does struggle to drive the lens's autofocus speeds to its full potential).
In the end, it's really up to what you use your camera for. If you use your camera just for animal ID (taking pictures of animals just for identification/record keeping), then you should get the x2 TC - it gets you more reach, gets the job done, and is cheap. However, if you want to take better and higher quality pictures, I suggest you invest/save for a new lens. That way, you can get much better autofocus, image quality, and wider apertures. Some great options to consider are the Sigma 150-600mm f5-6.3 and Nikkor 200-500mm f/5.6, but you should do your own research and think about what you actually need and what you don't.
When I had the chance, I decided to switch from the Nikkor 55-300mm to the Nikkor 200-500mm. I still don't regret my choice, and within the limits of the D5300, the lens just gives me more breathing room for me to be able to take great pictures.