I can recommend the free MatrixRate Table Rate Shipping extension: https://marketplace.magento.com/webshopapps-module-matrixrate.html
It gives you some more options than the default table rates but these options are covering a lot more use cases. Besides that the extension is used just like the built in table rates.
In your case I would use its weight vs destination setting which will allow you to upload a CSV with the following fields:
Country,Region/State,City,"Zip/Postal Code From","Zip/Postal Code To",Weight>,Weight<=,"Shipping Price","Shipping Method"
You now can state an exact weight range with not only a price but also a custom "Shipping Method" names. Customers see all prices/method names depending on their total cart weight in a list. Personally I'd advice to hide
.table-checkout-shipping-method .col-carrier via CSS as this will always show the generic name you gave the MatrixRate Table Rate option in the admin backend and my confuse customers if you make use of the "Shipping Method" field in the CSV.
I just installed this module in 2.1.8 installation and for me it worked out of the box (I did a manual installation).