"Should I mention this to the new buyer?"

Legally no. Morally no. By your standards may be necessary.

I wouldn't attempt to fix it at the stage. I doubt a plumber can dependiably test it,you certainly can't. Even if I knew the trap was leaking 1 oz per day I wouldn't bust concrete to repair it.

It's not uncommon for the water in an unused trap to evaporate, causing the trap to open. A trap opening up in a week is unusual, but there could be issues with the size or the type of trap used. Some people will pour a bit of RV antifreeze (RV antifreeze, NOT typical car radiator antifreeze) into the trap to prevent loss from evaporation. RV antifreeze is a "plumbing antifreeze" versus a "car engine" antifreeze, it's use is fairly common for traps in vacation homes, etc. RV antifreeze will not harm a septic system.

I'd go with evaporation as the explanation for now, I'd not feel compelled to put that on a disclosure. If you do the RV antifreeze and it opens up in a week, then it could be a leak. Only if I KNEW it was a leak, confirmed by a plumber, would I put it on the disclosure.

Unless you've got some sort of weird heat and airflow issue over that vent, no way it should evaporate that much in only a week. If you choose to try RV antifreeze, ensure it is a formulation that uses propylene glycol, not ethanol.

Another effective solution is pour a cup of bleach into the drain about every other week, then add another cup or two of water to ensure the trap is full. The bleach solution will break down bacteria and other gunk that can hold odors, leaving you with a relatively clean plastic or metal surface. It is a simple solution, but often very effective in preventing odors from a seldom-used trap.

Bruce

Has anyone ever used a product like this? Could this work for my situation?

https://www.supplyhouse.com/SureSeal-97042-3-SureSeal-Inline-Floor-Drain-Trap-Seal-SS3009V?gclid=Cj0KCQiAtJbTBRDkARIsAIA0a5MIyESkdGlDE8fGWz4XSuCY0h0rZfQMwtHVodNe0fMAtvSnX27gPm0aAscLEALw_wcB

If that fit's tight in your drain,it will prevent odors coming from drain regardless if trap is dry or not. But if something in basement leak's it has no where to go and will flood basement. You decide if that fit's your situation.

I think that product allows water to flow down but prevents the smell coming up.

Its just really odd. It has to be a venting issue. We would go on vacation or sometimes it would go for weeks with no issue.. Then sometimes we have to run water like once ever 4 days down the drain. Im just trying to figure out what we are doing to cause it to dry out quickly.. If we do wash more often, or flush the toilet in the same bathroom as the shower more often.. But im not sure how we will be able to figure this out.ugh

I would think if its a leak it would be consistent like like we would have to fill it exactly every 5 days or something. But its not.

"I think that product allows water to flow down but prevents the smell coming up."

Sound's like problem solved,noone can give you the why unless you furnish detailed drawings of sewer system.