Water-based doesn't necessarily mean water-soluble. Water is just the solvent used to carry the finish. Once the finish cures, the water-solubility is usually gone.
For example, consider Timbermate Wood Filler. Per the product description,
...Timbermate may react with low grade steels and turn black. If this happens, simply add water and remove it, or wait for it to dry and sand it off.
Basically, while the product is still wet (uncured), it is water-soluble. Afterwards, it must be sanded off, which to me means that water will no longer dissolve it.
Therefore, finishing with tung oil really won't matter in terms of water resistance since the filler is resistant to water once cured anyway.
You can test the water-fastness of your dry filler on a scrap piece to be sure if it will last over time. Just drill some holes on a scrap, fill with filler, let cure, and soak for a few days. You should be able to tell if the filler holds up. A cutting board shouldn't be soaked anyway, so this type of test should be sufficient to test the filler.
the board will be finished with Tung oil
Also, I would avoid tung oil for finishing a cutting board. Please have a look at this thread for further discussion on why it's a bad idea. I'm not certain of the food-safety of pure tung oil, but most products sold as tung oil nowadays have very little actual tung oil in them and are instead mostly polyurethane.
Most people just use food-grade mineral oil and renew it every so often. My previous statement about the finish not affecting the filler's water-resistance still stands with mineral oil.