How can I dispose of chlorine and sodium hydroxide as table salt and water? If I used chloralkali electrolysis of brine to produce hydrogen, and can't use the other products (or not all of them), what route(s) can I take to revert them back to brine? I figured, since I have to put in energy to take the brine apart, it should have some desire to reform.

With the $\ce{H2}$ taken away, the remaining products are:

$$\ce{2 NaOH (aq) + Cl2}$$

If I just mix them, I get sodium chlorate or hypochlorite. Can I process those further?

Both $\ce{H2O}$ and $\ce{NaCl}$ have a negative $\Delta H_\mathrm{f}^\circ$ ($\pu{-285.8 kJ/mol}$ and $\pu{-407.0 kJ/mol}$ respectively), and $\ce{Cl2}$ is already in its standard state, so we would need to overcome the $\Delta H_\mathrm{f}^\circ$ of the $\ce{NaOH}$, which is negative as well ($\pu{-470.1 kJ/mol}$). Can I use ion filters to some benefit?

Routes that allow retrieval of some of the energy spent for the electrolysis are favored (i.e. the more exothermic, the better).
It looks as if the simplest way would be to mix the products, and let the mixture warm up, so we get sodium chlorate in water. We can then have that crystalize and heat it until it disintegrates into $\ce{NaCl}$ and $\ce{O2}$. That would need no additional substances and have a clean end result. The only remaining question would be the energy budget, but the forming of the chlorate is exothermic and additional heat might be available.

Alternatively, we might find something useful to oxidize with the $\ce{NaClO3}$.