If the computer doesnt exist it shouldnt 404, it should return:
<BESAPI xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="BESAPI.xsd">
<Query Resource="(ids of it) of bes computers whose (name of it as string as uppercase starts with "not exists")">
<Result/>
<Evaluation>
<Time>0.574ms</Time>
<Plurality>Plural</Plurality>
</Evaluation>
</Query>
</BESAPI>
Im wondering a character in the computer name may be causing an issue with your query string