This is an interesting scenario. The FCC would likely take action against the station.
I would wonder if the FCC sees an Indian Reservation operated non-commercial station on tribal land as being beyond FCC jurisdiction. I do not know of any test case but I would think the FCC would err in favor of the station whenever possible regardless of whether on tribal land or not. Still, an interesting question/theory. I recall an FCC agent telling me when he visited a pirate the person did not allow entrance and stated they answered to a higher power, God. He left. No argument there but he sought assistance. Federal Marshals thought otherwise and I need not explain who won.
In the past, non-commercial FMs that run old radio shows from decades ago do not have to cut out the commercial content, likely because the commercial was a part of the show itself and the product was not a legitimate current day advertising offer. For example, a product advertised on a 1940s radio show was seen as okay.
On the other hand, commercials are covered by non-commercial stations that carry, say, network newscasts. Locally inserted PSAs cover the commercials. Since this is a very common practice, the airing of a syndicated show and allowing the commercials to air would not be viewed in a positive light by the FCC even if the station received no direct compensation for the commercials airing. Typically commercial programming is bartered for clearing the program's commercials, so technically since a non-commercial station is just that, the commercials pay for the station to get the program free. The FCC could argue a non-commercial station would have to pay a fee if the commercials were not aired, thus compensation occurred. Networks typically charge a minimal fee to be an affiliate when that station is non-commercial since the network benefits monetarily by stations airing/clearing the commercials.
Some believe if there is no monetary consideration for a commercial it can be aired on a non-commercial station. I'm sure the legal department at the FCC can punch many holes in that thinking.
Granted, if this was an isolated case, I believe it would be easily forgiven, if needed (ie: someone messed up and it wasn't their intent). For example, I was listening to the TCU non-commercial FM carrying the University's live sports feed when someone forgot to cover the commercial break with PSAs or Underwriting. The second break was covered, so this was a mere oversight versus a pattern of non-compliance.
I have always found staying away from that line in the sand is a good deal with the FCC. When you can demonstrate you want to be compliant, in fact more than just compliant, those instances where you weren't are easier for the FCC to forgive or at least only throw a chapter at you instead of the whole book.