Although the issue is somewhat contentious as Bill has stated, the bottom line is that your Health Dept seems to have made up its mind on the topic. IMHO the separate system removes the question of negative impacts to the septic system and also removes any possibility of cross contamination.
It should not be too expensive (I know, subjective) to install this drain, a small plastic "drywell" with no engineering involved should be sufficient, but see what the health dept recommends for products and location.
Id be interested to know if this was a homeowner installation, because Id like to think if you dealt with professional salesmen and installers they would have known the requirements of the local health dept.