We have a cabinet shop and finish alot of wood. You have a few options, Depending on the wood and if the table has a solid top and you can take it off take it to someone with a big drum sander and have them take it down raw and refinish it, sounds like alot of work but really not. If you can take it off and want to travel I will do it for you, just make sure that you bring all the leaves with it so they are all the same thickness.
If the top is veneer then sanding is out, if it was finished with a pre-cat laquer you can sand it lightly with 200 grit and respray the entire top. If it was finished with a conversion varnish you could could re apply it to the spot and but would have to feather the edges with some laquer thinner, but it would still look flawed.
But by far the center piece idea would be the easiest.