Generally, no. Where Miscanthus is invasive, it propagates by self seeding, and its roots shouldn't bother your septic system. If anything, the soil around the septic system may be too rich for Miscanthus to thrive.

A couple of questions:

What zone are you in? This is important to know, because at the bottom of its hardiness range, Miscanthus doesn't self seed as much as at the warmer end.

How moist is the soil in the immediate area? The boggier it is, the better the chance the Miscanthus may self seed.

Thx for the info.
We are in zone 5 and our cottage is located on an old gravel pit from years gone so any water drains quickly.

Zone 5 is at or near the bottom of the hardiness of most popular cultivars. It may never develop ripe and viable seed. I wouldn't worry about invasiveness.