Refusing to enable another (apparently strong despite its size) nation's lifeline while on a conquest of the entire territory they need to perpetuate said lifeline?..
Sound like a prime method to make them insert themselves into your conflict to enable your opponents to resist or even strike back!
I can only imagine they will be sending supplies, elite troops, instructors, and technology to the ones resisting in their favour. Maybe not enough to tip the tides. But absolutely enough to make that decision of refusing their trade to cost them more than whatever they gained from it (unless it is explicitly a "not while still fighting, to avoid selling intel and to our foes" type of deal).