Please give us references, examples, texts, contexts, or indeed any information about the verb 'bestir', and how it might be used.
There is in theory a possible overlap of use, in some very limited contexts.
Of course I know the verb 'bestir (yourselves)', but I doubt I have ever used it.
Well, in truth this is a general inquiry into the differnce between be- prefixes and the root word itself, when they seems to have much the same meaning/ function (unlike belike vs. like, or beware vs wary for example).
Stir yourself, man! (bestir yourself, man!)
He was much loved by all. (He was much beloved by all)