I have read several very clever propositions for how to count in a dozenal arithmetic system. But, I fear that these systems are too clever, and not practical for creating a smooth transition. There really is no need for a new verbal numbering system in order to apply dozenal arithmetic. The words "ten" and "eleven" aren't really decimal, they don't fit in the "x-teen" pattern, and will serve quite well in dozenal arithmetic.
For "12", a.k.a. 10 in dozenal arithmetic, the word "dozen" will serve. That is, one dozen and one, one dozen and two, etc. all the way up to "eleven dozen and eleven". At which point, we use the already existing word "gross". "one gross","one gross and one", all the way up to "eleven gross, eleven dozen, and eleven".
At this point, a new word is needed, to signify a dozen gross. I would suggest a "cube". Twelve singles line up to make a row of a dozen (10), twelve dozen line up against each other to make a "square" of a gross (100), and a dozen gross stack to make the shape of a cube (1000.) This would be easily visualized and understood by someone new to the system.
After this, it might be easy to mimic the decimal practice of using "ten thousand", "a hundred thousand" etc. You'd have "one dozen cubes" (10,000) and "one gross cubes" (100,000), going up to "eleven gross cubes, eleven dozen cubes, eleven cubes, eleven gross, eleven dozen and eleven". Or perhaps "eleven gross, eleven dozen and eleven cubes, eleven gross eleven dozen and eleven". This looks awkward, written out, but no more so than "nine hundred and ninety nine thousand, nine hundred and ninety nine".
We would then need new words only for every third place, as numbers continue to get larger. But these would be used relatively rarely in daily life, and would not impact as directly on the daily use of numbers and counting.
There are several advantages to this type of verbal numbering. First, it is instantly familiar. Quite a few people use it informally without even realizing that they are already using dozenal arithmetic, (e.g. I bought two dozen eggs). Almost anyone can mentally grasp what the numbers signify.
While numerals are tools in math, numbers are tools of language. They need to have instant meaning to both the speaker and listener. Someone, today, who talked about a gross or a dozen, instead of a hundred or ten, would be instantly understood, perhaps seen as quirky, but quite comprehensible in their choice of how to express numbers.
If dozenal arithmetic is to be a serious alternative, rather than a hobby for mathematicians and idealists, it needs to be kept simple, with the smoothest possible transition for the typical user. The deepest fear of dozenal math is the fear that a novice user won't know what the numbers mean, or how to apply them in daily life. Sticking with the words "ten", "eleven", "dozen" and "gross" would go a long way in eliminating that fear, and making the transition practical.