You did well to avoid the clips weeal!
cazz is right, spiders travel around the house, usually unseen, but when they encounter a bath or basin, they can't grip to climb out, which is why you find them there - hence the myth that they crawl up drinpipes. Apart from the fact that there are svereal inches of water in your basin drain trap which a spider could not get through, they are far too clean to go anywhere near a wet mucky drainpipe - unless it's a oneway trip out of your bathroom!
If you can stand it, drape a towel over the edge of the bath touching the bottom, and leave it overnight. Mr Spider will be gone in the morning, and you'll probably never see him again. They do good work killing disease-carrying flies, so try to be nice to them.