No, you should get the 55-250.
55-250 is a goal to reframe. 70-300 is a full frame goal. The full frame lens gives you a slight advantage in terms of focal length, but I would not say that 300mm will allow you to take pictures that 250mm do not give you. 300 mm is only 20% more than 250 mm.
The reason I suggest 55-250 is that 70-300 is optimized for a complete image. For a large model camera, the image quality should be similar to that of 55 to 250 pixels: you will only see benefits in terms of image quality if you use the image circle complete. In addition, the 70-300 because of its large image circle is significantly heavier and much more expensive. You pay for heavy glass that you do not really need. In contrast, the 55-250 circle has an optimized picture circle for a compact camera, so there is no unnecessary glass.
If you really think you need more focal distance and are willing to pay a lot of money, consider a 100-400 lens: Canon, Tamron, etc. unnecessarily heavy on a camera. In addition, the Canon probably far exceeds your budget (which you did not specify, by the way), while the Tamron is significantly cheaper, but still costs several times the price of a 55-250 .
I would go with the 55-250 given your camera potential. It is a very good goal for a very good price and a very good weight. It has its limits, but they mostly come from its picture circle of the size of a crop, and you can not overcome it anyway with a camera.
However, that said, if you take very fast moving animal pictures such as birds, the faster autofocus of 70-300 might present some advantages.
Edit: I repeat the statement that 70-300 has a similar image quality. In reality, he is poorer on a camera! See here for comparison.