How to paraphrase distribute evenly possible

Distribute as evenly as possible means "distribute as evenly as (is) possible", in other words, as evenly as one can. It is a comparative construction. The commanded distribution is to be compared with a distribution evincing the greatest evenness that can be.

"evenly possible" makes no sense as possibility is neither even nor uneven, regular or irregular.

Compare:

Jump as far as possible.

Jump far possible ^{ungrammatical}

In America people might say “let’s divy up the stuff” or depending on how many people are involved they might say “let’s split this stuff 2/3/4/etc ways”