I know Cl- is a pretty harsh oxidant, so would the chlorine be oxidizing the ammonia present from the bacteria into nitrite, then to nitrate, which would elevate these levels? Is there currently a way to remove nitrate from the pool without doing a drain/refill? I know there are nitrate removers for fish tanks, but they use polycaprolactone as a carbon source for the denitrifying bacteria in the tank which wouldn't (shouldn't) exist in the pool.
Cl- is chloride ion and is not a harsh oxidant so did you mean HOCl which is hypochlorous acid? The latter is an oxidizer, as is OCl- hypochlorite ion. These are selective oxidizers and which one is stronger depends on the chemical species being oxidized. In the case of ammonia, it is HOCl that combines with ammonia (NH3) to form monochloramine and does further substitutions to form dichloramine or nitrogen trichloride, but these are substitution reactions (Cl+ replacing H+) and not oxidation. HOCl can then oxidize some of these to form nitrogen gas or nitrate. Mostly though, the largest final product is nitrogen gas. The proportion of nitrate from the reactions is usually around 10-20%. This spreadsheet can be used to calculate what happens using the Jafvert & Valentine and other models. See Chloramines and FC/CYA and you might also be interested in Oxidation of Urea as well.
There is no easy way to remove nitrate from pools other than by water dilution. However, nitrate is not a problem in pools (why do you think that it is?). It is an essential nutrient for algae just as with phosphates, but that is irrelevant if one maintains the appropriate FC/CYA ratio which has chlorine kill algae faster than it can grow regardless of algae nutrient level. Algae is ultimately limited in its growth rate by sunlight and temperature.