Save your money for shampoo, take the trap off , pull the `wig' out and replace the trap.
Do NOT pour water down t`ole with the trap off.
Not the most pleasant job in the world but better than chemical uncloggers.
At the risk of sounding stupid (which my kids say I am) - I assuming to reach this you take the side of the bath off and attack from beneath 8doubl;es? My dear OH is phobic about most DIY and would probably call a plumber in, but I don't mind having a go!
Yes, put the plug in the bath to stop any water , bath panel off, paint tray or seed tray wiggled under the trap to catch the water,
undo trap from bath and waste pipe then you can put the trap in the tray and pull the tray out without making a mess.
Take care not to lose the rubber seal from the bath connection the other end should stay on the pipe.
Re-fit (careful not to cross thread) as tight as poss by hand or a gentle tweak with waterpump pliers.
Put a 4" of water in the bath and pull the plug and check for drips giving the joint another tighten if needed.Bath panel back on. Job done.
Always have a couple of old towels handy when doing plumbing jobs.