Disclaimer: I have very little practical experience with clay.
It will probably be easier and more practical to remove the current bottom entirely and replace it with an undamaged one.
The main reason is that if the bottom already droops now, you'll have a really hard time reinforcing the bottom without accidentally destroying the entire vase. To create a strong bond you have to kind of massage the new clay into the existing one, otherwise the bond is prone to cracking during the drying or firing phase. That means supporting the damaged bottom from the inside of the vase without accidentally damaging even more of it.
It will be easier to create a strong bond around the rim of the new bottom than over the entire surface of the old one. You can even give the existing walls of the vase some additional drying time (but not too dry!) to make them more resilient against bending and drooping.
And since you already know that the bottom likes to stick to the surface, intentionally make your new bottom thicker and then cut it to the desired thickness when you cut the vase from the table. You can also leave the bottom a bit thicker than desired and then cut it to it's intended thickness after some drying time. I suggest watching some videos of people forming clay dishes to see this precise shaping of a half-dry object. It's far easier than trying to give very wet and malleable clay it's final shape and then getting it to keep that shape without deforming.