The problem is that you hear a mix of your acoustic sound and the amplified sound when you play but only the amplified sound out of the looper.
The best solution to it is using an EUB, so your sound is always the pickup sound only, directly or out of the looper.
The next best solution is to try to come close to your acoustic sound out of the amp, which is really difficult. I came close to that with a Full Circle, impedance buffer (might not be needed) and a AI Ten2 a few days ago when we tested some amps with my bass (I haven't tried bowing in this combination).
A microphone would deliver a better sound but will probably pick up the looper output (and other sounds that should not be looped), so it might only work in solo parts.