It says that the maximum continuous current is 20A. That 38A controller will exceed that, probably for most of the time. The BMS doesn't cut off until 60A, so you should be OK from that point of view, but I would think that you'll probably get cut-outs due to voltage sag if you run at full power for too long. Basically, You'd be better off with a 22 A controller, which I think is what's supplied with that kit. Also, if you're going to buy a new controller, one with an LCD would be better because that will allow you to use the pedal sensor.
Here's a good 22 amp controller without LCD, so good for throttle only:
S-KU93 Controller. It is suitable for all our 250W, 500W motor and Q128 Motor. This controller is compatible with 48V and 36V battery.
bmsbattery.com
This is a good sine wave one with LCD. It's 30A, but you can adjust the max current in the settings. This one will allow you to tune the current to what your battery is happy with. You have to order the LCD separately from them.
S12S 500Watts Torque Simulation Sine Wave Controller for 36Volts and 48Volts battery pack E-Bike System
bmsbattery.com
The controller is the most important part of an ebike. It determines how well the motor and the pedal assistance works. I think it's worth spending the extra money to get a decent one.
Your motor can handle the 38 amps, no problem. It would also go very well at 25 amps and 48V.
I'm pretty sure that your battery isn't really up to the job even with the 22A controller. it'll work, but range will be poor and I don't think it will last long, so you'll probably be looking for a decent 48v battery soon. Bear that in mind when you decide on what to do about the controller.
Another thing you could do is get the Yose Power kit with the sine wave controller and the smaller motor. That'll cost about £200, and your battery will be much happier with it and last a lot longer. It'll be much more efficient, more pleasant to ride, and it'll go up hills better.