Will brushless controller of Current: 17±1A work on a 13A battery?
17A+/-1 is not valid without conditions.  

This may refer to the surge current for starting a motor which can take 5~8x the Watt/Volt ratio but it affect internal temp rise affect by initial conditions and ambient T.  

Batteries are rated in storage time in Amp-hours [Ah] and Cranking Amps CA so 13A must be Ah.... 

Hopefully it represents a current limited switch sensed and controlled by PWM. with a threshold tolerance (if true)  17A compared to your motor 250w/50V =5A is roughly  17/5A= 3.6x rated load should give good acceleration.

15Ah implies you can go full load 5A for 3h with no acceleration or less with accel.

FWIW, Ebay specs are almost non-existent  and not worthy of trust, but it may be ok for you. Ask for an e-file of user manual.