What you are suggesting is akin to trying to measure your own motion across the Earths surface in, say, 1 hour, against your motion due to
plate tectonics, using a perfectly elastic ruler you can never be sure you aren't stretching, whose increments are too large to measure the motion of plate tectonics!
And as for the CMBR "rest frame", it is a convenient frame to use in cosmology, when dealing with distances and ages the span billions of years, but a real clock can never actually remain in that frame (gravity again).
The Solar System is moving at something around 600 km/s or so relative to the CMBR rest frame (the error bars for this estimate are quite large and it took years of analysis of the WMAP data to get this figure), but there is also a lot of gravity in these here parts. These are not calculations that can be done "on the fly", and even if they could they would still be subject to the other problems we have mentioned.