I think I see your point Dave in two possible ways (there may be more):

'as best (only) he can' (where 'only' is implied)
or
'as BEST he can' (with emphasis on 'best' - perhaps by means of a short pause, as if there was a comma between 'best' and 'he' - although perhaps still ambiguous)

However, although this is something very common in Greek as well as Italian and Spanish (i.e. emphasizing by changing the tone of the voice and/or pausing), my experience living in the UK for 15 years has lead me to the conclusion that such emphasis is often not heard and subsequently not understood by many native English speakers (but not all).

Nonetheless, as my posting name implies, I am Greek, and English is NOT my mother tongue. I would, therefore, like to ask the following:
If I wanted to WRITE "do {something} in the best way you can" (ignoring the fact that it may be perceived as patronizing, since no-one would expect to be asked to do something sloppily), what would be the best way to convey this in writing?
'as best as you can' ?
'as best you can' ?
'as well as you can' ?
{something else} ?
Or, perhaps, do away with the whole thing and simply write 'in the best way you can'?