We have the same type setup, gets plugged up about once a month or so. Wife just pours in some of the strong fluid available in Tescoe etc, bright yellow plastic container. Let is set about 1/2 hour then run some water. No real hassle.

soap and toothpaste make a lot of gluey gunk. mixed in: some of your partner's super - strong black hairs.

the reason i was asking where the pipe drains to is: only sewer - plumbed connections need 's - traps'. they are to prevent foul gas to emit back up through your plumbing.

so, solution #1: eliminate the 'smart - trap' and replace it with a proper 's' trap.

or

#2: do away with the trap altogether and put up with the stink...

you can keep the basin plug in to alleviate.

a lot of free advice for you to contemplate - or ignore.

I've done a bit of plumbing here, the residue of toothpaste and soap plus the hair bits can take a bit of moving! There is no real reason for the trap, it doesn't keep the smells out and its only function is to stop the pipe from blocking up further down the system. Better to remove the trap altogether and replace it with a 90 degree elbow. Chock dee!

It keeps the smells out - that is the reason the vertical pipe is cut below the water level (horizontal pipe).

Be careful about chemicals. THey may solve the immediate problem but soon you may find they killed the bacteria in the tank that process the gunk. THen you have to get the tank pumped out which you should never have to do.