Among my Arab friends with that name the spelling that omits the 'c' is more common. Another common form is Ahmad which is still pronounced the same.
The version with 'c' is one that contains a pronunciation hint for people whose native language is not Arabic (but probably English). As is the one with the 'e' vs the 'a' as last vowel.
I.e. Ahmad == Ahmed == Achmed.
What hint would that be? There's no 'c' sound in the Arabic version.
It is indeed closer in terms of its place of articulation to English 'h' than either variant of the German 'ch' sound.
To my ears [ħ] sounds closer to [x] and [χ] than to [h] (even though the place of articulation is closer to [h]), but I'm sure it's different for people who (natively) speak a language with all three.
... yes, this is a joke.