Respectfully: It is only a minhag not ride a bicycle (accepted by much of khal yisrael) with in an eruv. Several HaChamin allow it including the Ben Ish Chai who wrote that it would even be preferable to not doing a mitzvah on shabbat like visiting the sick or parents. Although it should said that this meant to be limited.
Shalom, Thank you for your reply to our ruling about bicycle use on Shabbat. What we wrote - that it is forbidden to use a bicycle on Shabbat - is in line with all major codifiers of today. Whilst it is true that the Ben Ish Chai (in his Rav Pelaim) ruled that one may ride a bicycle in Baghdad, one should note firstly that this was only in a walled city, within the eruv. Secondly, the Kaf HaChaim (404) argues against this ruling. Then Rav Ovadia Hadayah, in his Yaskil Avdi, who also rules that it is forbidden, suggests that the Ben Ish Chai changed his ruling later in life and forbade bicycles on Shabbat even within the eruv. (Though there is room to argue whether this retraction is universally acknowledged). Both HaRav Ovadia Yosef (see Yalkut Yosef, Shabbat volume 5, page 56) and Shmirat Shabbat KeHilchata rule that one should not ride bicycles on Shabbat. When we stated that "one is not allowed to travel by car or bicycle" we reflected the standard ruling of Rabbis today, based on halacha, custom and acceptable practice. Blessings.