I recommend contacting Burris for a compatible remote cable switch for the Eliminator III.
The Burris Eliminator series has 5 generations. The first generation was push button only for ranging. The second generation had a bluetooth remote for ranging activation in addition to the push button on the side of the scope. For whatever reason, Burris did not do a bluetooth remote for the third generation, instead opting for a WIRED remote pressure switch that you would connect to the scope and velcro to the side of the stock which people found clumsy and prone to snagging. Consumer feedback brought the bluetooth remote back for generations IV and V, with the remaining generation IIIs having a hard time selling until Fall of 2023 when vendors finally discounted them so much to get rid of them (50% off) that the new old stocks finally got depleted. Therefore, there is a very good chance you can get the WIRED pressure button switch from Burris, but just do not expect it to be a bluetooth wireless option as Burris never had that as an option as it did for generations II, IV, and V.