The first thing you need to do is to see whether you can remove the bathroom sink stopper by hand. If this is the case, then you will have saved yourself a lot of unnecessary work and stress.
To do this, take hold of the stopper with your fingers, and try and turn it. If you do not feel any give, or if there are no signs of movement (other than spinning on its axis), then your stopper will require more work.
This will involve going underneath the sink, and this is where the cleaning will come in handy. Remove anything from underneath the sink, and ensure you have enough room to work and maneuver.
The third step requires your flashlight. Use this to locate the horizontal plunger rod that connects the strap to the stopper itself. Once you have located this, then look for the rod that connects the vertical strap.
This shouldn’t be difficult to locate, as it is full of holes and fairly distinctive amongst the other components. You will then see a small metal clip that holds everything together.
The other end of the rod will be connected to a pivot nut. This will require unscrewing, and you can use the wrench to assist with this if it has become stuck or too tight.
If you have followed these steps correctly, then you should be able to release and remove the bathroom sink stopper from beneath the sink, and you should be able pull it out without any issues or resistance.