It will probably depend upon how securely you can mount and attach the guide scope and the barlow to each other and then those two as a unit to your main scope. Plus, when you use a 2X barlow you're going to be cutting your guide star signal by a factor of 4X (2^2) which will affect the signal-to-noise of the guide system. And, with a 2X barlow you're only going to have one quarter of the field that you had without the barlow and that will mean fewer guide stars from which to choose.
That said, if you are already guiding at a scale that is no more than 5X that of your main imaging scope and camera then you are probably okay with your current setup. An ASI224MC on a 120mm focal length guide scope will produce an image scale of 6.45 arc seconds per pixel which means you can probably use that without issues when imaging at 6.45 / 5 ≈ 1.3 arc seconds per pixel and greater.
I use a 120mm f/4 guide scope on my 0.8X-reduced Stellarvue SV80ST2 scope but I've paired that guide scope with an ASI290MM camera which has 2.9um pixels and thus my guiding scale is 4.98 arc seconds per pixel which would allow me to guide an imaging system that produced a 1 arc second per pixel image scale. So, I'm good to go and I often guide down below 0.5 arc seconds total RMS error using PhD on a A-P Mach1GTO mount. In this case, however, part of my success is having a good mount and a really solid mounting point for the guide scope. And, if the seeing isn't very good I can't stay below 0.5 arc second RMS and in that case a longer guide scope wouldn't help anyway.