After having consulted a number of resources   , I have concluded that there is currently no synonym of inching that is based on a metric word.
It takes time for such expressions to develop, and I'd be surprised if a phrase like "centimeter forward" ever evolved. But what do I know?
Anyway, until (and probably even after) that happens, using inching is perfectly fine. It doesn't matter whether you are somewhere where the metric system is used or not. You're not going to cause any space disasters merely by using a word.
If you really wanted to use a metric word you could say something like "advancing a centimeter at a time", but that seems rather inelegant to me. I'd stick with inching.
Like Max said, it's a figure of speech. For example, I might tell my friend not to put all his eggs in one basket. Hardly anyone transports eggs in baskets anymore, and certainly, my friend doesn't.That, however, doesn't prevent me from using that particular figure of speech as an effective way to emphasise a practical point to him.
How do I avoid using archaic measurements?
You could avoid referring to units of measurement at all. I have looked at the synonyms of inching, but in my opinion their negative connotations make most of them inferior to inching. (Provided, of course, that you do wish to avoid such negative connotations.)
I did find, however, one possible candidate:
To move slowly or with small movements in a specified direction.
This matches with the definition of inching from the same source:
To move very slowly or by a small amount in a specified direction or
Elon is edging his way towards Mars.
Please note that even this word may have an unsuitable slang connotation, though perhaps rare. In either case, I do find inching a better choice.