Can I route an irrigation main through my basement?
In most US jurisdictions, from a legal perspective, the work in the basement would fall under the local building code and its permitting, licensing, and inspection requirements...e.g. a splinkler contractor could not run the pipes inside the house and a plumber would be required, building department fees would be applicable, etc.

The pipe running through the basement that conceptually would be part of the irrigation system, is both legally and from a life safety standpoint part of the domestic potable water supply [it may be possible to get the building department to treat it as a domestic non-potable system, but that's less likely than a flat "no" from the building department]. Even if you get it approved, you'd still want a vacuum break where the pipe exits the building and there's a risk of contamination.

Anyway from a practical standpoint, one thing you're facing is the standard practices of the irrigation installer tradebase. They own Ditchwitches and shovels. They don't usually dress for walking across homeowner carpets and aren't in the habit of pulling plumbing permits. This is pushing things outside their business model...and reducing their revenue. Maybe you see a portion of whatever savings the contractor realizes, maybe the contractor just passes on the job [it depends on how much work they have, and you want the person who is busy].