Plumbing – Do i need a backflow preventer for drip irrigation system

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].

I will potentially be drinking the same water that flows back in the well after the frostless hydrant is shut off - yes?

No. Freezeless faucets drain their water back into the ground at their base below the frost line when turned off, NOT IN TO THE SUPPLY. The freezeless facuet drains this water so the pipes above the frost line have no water to freeze in them; hence a freezeless faucet.

Will a simple $5 plastic backflow preventer be all I need?

You get what you pay for. Will a 5$ part function as well as a 125$? Of course not! Will it work well out of the box? probably yes and probably fairly well. But it will not be as effective at stopping backflow, it will probably not work for as many years to come as a true backflow preventer, it probably cannot be tested, and it probably cannot be repaired.

...or is there a certain product meant just for this?

There are true backflow preveters for this that are designed to effectively stop backflor that are able to be checked and tested for reliability and repaired. These typically installed run 200-300 in parts alone for the backflow, fittings, boxes, etc. Labor can easily double or triple this in many cases.