Consistency in photo managing (Bridge/Lightroom Classic)

Yes, you are correct, the flags in Bridge are ignored in LRC and vise versa. This is well known for a long time and I doubt that Adobe will fix this any time in the future.

Why? The primary difference between Adobe Camera Raw in Bridge, and LRC is that the latter is a ACR with a database. There is NOTHING that you can do in ACR that you can't do in LRC. To go back and forth between the two applications is unnecessary and a waste of time. Plus, when you take a photo from LRC, make changes in ACR, and then later open the same file in LRC, you are hit with the question: There have been changes to this file outside of Lightroom, which version do you want? LRC or the other application. I'm going to guess that you select the latter here but there's no point.

I am sorry, it's always dangerous to point out that a workflow that someone is happy with is not a recommended workflow. If you think my words here are out of place, so be it. But in my first paragraph I let you know why it is and what to expect of your wish. But I also suggested ways to avoid the issue in the first place.

Another "more common than you'd think" workflow that occasionally creeps up in these forums are people who take the card from their camera and start to process the images ON THE CARD. No backups, no space on the card for processing, nothing. And then they get pissed off at folks trying to help them. 

Please try and just stay in LRC for your processing. Command/Control-e to work on an image in Photoshop, then save and close to get the updated file back into LRC as needed. Please, just try it.