I put in downspout drainage under a sidewalk at my house last fall.

4" corrugated drainage tile which is actually 4.75" OD so the tunnel under the sidewalk was much larger.
I just backfilled and soaked the dirt with a hose until it settled, backfilled again and soaked it again.

Another method of you're worried about supporting it is use expanding foam. It's a method used to Jack up sunken concrete driveway slabs, and a newer way to support fence posts. It will work just fine.

Personally I wouldn't really worry about it much. It's such a tiny area