Or you can say it is the conversion between a gram and a dalton, which is the same but more complex as it is now intertwined with the gram

I already said exactly that:

>>> Avogadro's number, and therefore the mole, is the conversion factor between atomic mass units and grams.

But that doesn't intertwine anything with grams. I went on to say

>>> Similarly, 3 is the conversion factor between feet and yards, but nobody thinks 3 is a fundamental base unit of the imperial system.

>>> It would be total nonsense to say that yards are defined by reference to 3.

It is a base unit, a unit for lots of stuff, and it is not dimensionless it is 6*10^23/mol

I'm pretty sure the mole is not defined as 6 * 10^23 mol^{-1}.

There is a concept of "the Avogadro constant", which is defined to have units of mol^{-1} (at least, according to a cited statement on wikipedia), but that is not a coherent concept -- since mol is dimensionless, mol^{-1} is also dimensionless.

Just look at https://en.wikipedia.org/wiki/Mole_(unit)#Criticism :

> Since its adoption into the International System of Units in 1971, numerous criticisms of the concept of the mole as a unit like the metre or the second have arisen:

> the number of molecules, etc. in a given amount of material is a fixed dimensionless quantity

> the mole is not a true metric (i.e. measuring) unit

Or look at https://en.wikipedia.org/wiki/Atomic_mass_unit :

> One unified atomic mass unit is approximately the mass of one nucleon (either a single proton or neutron) and is numerically equivalent to 1 g/mol.

amu and g are both units of mass, so 1 amu = 1 g/mol is an explicit statement that mol is dimensionless.

Calling mol a unit won't accomplish anything except corrupting your dimensional analysis. mol is not analogous to the SI units meter, second, ampere, gram, kelvin, etc. -- it is analogous to the SI prefixes kilo-, mega-, milli-, micro-, nano-, etc.