nowadays, it's likely that he can search it even if removed from the vehicle. when you're lawfully arrested, the officer will get to search everything on your person both under "search incident to arrest" and "inventory search". depending on the circumstances of the arrest (which we don't have here), the officer is often going to have probable cause to search the bag and the rest of the car. if the officer does not, he'll usually search the car anyways...

• if he finds nothing, then it doesn't matter. a defendant generally doesn't sue for a simple civil rights violation like that because then it forces the defendant to testify (defendants almost never testify). as a result, the only remedy is usually exclusion of evidence in the criminal trial, and in this case we said he found nothing, so there's nothing to suppress.
• if he does find something and thinks his ground for probable cause aren't 1000% clear, he will usually lie to backtrack on getting probable cause. so for example, if he finds pot, he'll say "i smelled pot when i pulled them over". other times he'll flat out lie and say the defendant said something they didn't say, or that the drugs weren't actually in the bag... they were on the floor in the back seat in plain sight (another exception to warrant requirements). the defendant is almost never going to testify, so the officer's statements will be taken as gospel. for those who don't think this happens, this got outted as widespread practice in the NYPD discrimination lawsuits.

there's also the question of how the bag got outside the car. if a defendant is saying "the bag isn't mine" then the defendant doesn't have a 4th amendment right against that bag's search and seizure because no one has a 4th amendment right against search and seizure of property and privacy that's not their own. i already said that if the defendant keeps the bag on their person, it'll be searched under SIA and inventory. but if the defendant pitches the bag, that just looks suspicious, even if they admit that it's theirs. there's probably case law on that, but i never encountered it, but my money would be on probable cause to search the bag.

[also a lawyer]

moral of the story: DO NOT rely on warrant requirements re search/seizure to save your ass.