Almost a decade ago, I was in a trip to the USA. I was very surprised to see Liter used for car engine sizes (here they usually use cubic centimeters, which are exactly 1/1000, so not a real problem).
I remember strolling through a supermarket, and looking at the soda bottles, which were bigger than the 1.5L bottles common here. I picked one up to see what size it was. I guess most readers know the answer - it was 2 liters. I remember wondering how come Americans are willing to use a metric unit.
So I asked a vendor. His answer was "This isn't a metric unit. It's liter".
So I asked him how much was a liter, and his answer was something along the lines of "33.8 ounces" (without blinking of stopping to think about it).
Which, of course, got me my answer. The reason Americans are using a metric unit is because they don't know it's metric.
The problem with your proposal is that, if implemented that way, means just adding another unit to the mix, without exposing people to the main advantage that the metric system has to offer. That does not bode well for a "migration path".
You should add to that the fact that volume realization is hard. I'll give a couple of examples. First, bear in mind that the two units people are, more or less, familiar with are a milliliter (1 cubic centimeter = 1/1000 of a liter) and a liter.
The first was when a company I worked for ordered a certain amount of boxes for their product. We were trying to figure out whether we have where to store them. I made the calculation, based on box size, and figured the entire bunch would require a little less than 2 cubic meters (around 1.8). We sort of made a hand gesture estimate and figured it was not that much. Boy, were we wrong. We ended up using up every spare cabinet and space in the office. Lesson learned: a cubic meter is a lot.
The happened just yesterday. I was telling my wife we will have to remove some soil from our garden to make space for extra flooring. She said "we'll be giving that to friends, right?". I told her it was about 500 liters of soil. I then made a quick mental calculation. We'll need about 12 squared meter of flooring, and the base is about half a meter deep. 6000 liters. Assuming soil is half as dense than water (it was while driving, so I couldn't look it up), you get 3 tons of soil. My instinct was off by a factor of 10, and her instinct was off by a factor of 1000.
This doesn't mean this is useless. Can you make this same calculation, off the top of your head, using imperial units? Metric does simplify things quite considerably. It's just that, specifically for volumes, that is a hard problem to solve.