It is apparent, that the name "Bedichek" was anglicized from the original Czech name"Bezdicek" which is a common name in Bohemia and in the Czech Republic, generally. When you include the characteristic Czech diacritical marks, then the name looks like this: "Bezdíček". The accent over "i" makes it sound long, like the English word "deal".The hook over the "c" makes it soft and reads like "ch" . That's probably why your ancestor changed it to "ch", to make it sound like the original Czech name. As for the missing "z", he probably wanted to make his name appear simpler.
Roy Bedichek wrote in 1934:
"Below is a verbatim copy of information my father (James Madison Bedichek) wrote down for me about 40 years ago, when his memory was still fresh:
signed -- Roy Bedichek: -- (and dated August 31, 1934)
'John Joseph Bedichek was born near Prague about the year 1759. At an early age he enlisted in the Austrian Army and was in many battles against the Turks. He had seven horses shot from under him and was wounded four times. He served 10 years, and his command being stationed on the Rhine, sympathizing with the Prussians against whom he was now fighting, he, and two others swam across the Rhine into Switzerland.'
'He was an independent thinker for he was a Protestant among Catholics. He could read and write among an ignorant populace. He wooed and won the heart of Miss Elizabeth Ross who required of him to get official testimony from his family standing in his native country; this proving satisfactory to her they married in 1799 in Canton Doveau, Switzerland. He died in 1815.'
'They had two sons and three daughters, John Joseph, Frederick Augustus, born April 9, 1809, Milani, _____________, ___________ of whom Malani was the oldest child and Frederick the youngest of the two boys.'
'The older brother enlisted in the Imperial Guard of France, being the proper height, 6 ft., 6 in. He soon became an officer and was sent home to recruit for that noted Swiss Regiment. Not many years after he died of smallpox.'
'Frederick Augustus learned the cabinet trade and carried on a large shop in Bern, but afterward came to New York in 1837. While there he made the costly furniture for the St. Charles Hotel in New Orleans.'
'He remained in New York four years, but the hard times caused him much loss. He went from there to Virginia. He married at Ripley, Virginia a Miss Mahulda Jeviden during the year of 1849. He moved to Buffalo, Virginia and engaged in the furniture business. Years afterward, having nearly all his property destroyed by fire, he moved to Johnson County, Missouri in 1857.'
'Malani married a man by the name of Smither and moved to the United States and settled at Buffalo, Virgiinia.'"