Hi ! It is possible that the blood which came while spitting and brushing the teeth could be due to a trivial injury
caused by the broken plastic. However, I do not consider that this small piece could cause the type of discomfort you have described for your abdomen and back. The small piece which you might have swallowed will with all probabilities come out in your stool matter, and you may not even notice it.
The discomfort in your abdomen could be just co incidental which naturally have been attributed to the ingestion of the plastic. However, if the symptoms persist, then you should visit your family physician, who may advise you for an ultrasound
abdomen for further needful.
Wishing you all the best. Thanks.