I don't know if this is the person you are looking for, but I found a birth certificate for a René Rachel Theophiel Coudyzer, born on 1887-03-14 in Harelbeke, son of Theophile Coudyzer (age 33) and Elodie Coucke (age 27).https://familysearch.org/pal:/MM9.3.1/TH-1942-21075-57593-85...
(ten year table index)https://familysearch.org/pal:/MM9.3.1/TH-1951-21088-34505-61...
(detail, entry #103)
Theophilus Coudyzer and Elodie Coucke were married on 1883-06-13 in Harelbeke. Here are links to their marriage certificate.https://familysearch.org/pal:/MM9.3.1/TH-1942-21076-2426-65?...
(ten year table index)https://familysearch.org/pal:/MM9.3.1/TH-1951-21088-36271-68...
(detail, entry #178 which continues onto the next page)
The marriage certificate cites both the parents and grandparents of the groom, but as far as I could tell, only the parents of the bride (I've done my best on all spellings/transcriptions but it's best if you confirm them all yourself).
Theophilus is cited as being born in Waregem 1854-03-03, son of Gregorius Coudyzer who died in Waregem 1872-01-15 and Melanie Bekaert who died in Beveren bij Harelbeke (a village in between Harelbeke and Waregem) 1882-08-17.
Gregorius's parents are cited as Ambrosius Coudyser (yes, it does appear to be spelled different than all the other spellings) who died in Waregem 1863-05-03 and Maria Josepha Waelkens who died in Waregem 1827-12-29.
Melanie's parents are cited as Franciscus Bekaert who died in Waregem 1845-04-02 and Maria Theresia Tronquay who died in Gent 1854-09-05 (in order to confirm the spelling, I found her death certificate https://familysearch.org/pal:/MM9.3.1/TH-266-12873-102553-57...
Eulodie Coucke is cited as being born in Harelbeke 1859-11-24 and the widow of Julius Jean Coudyzer who died in Harelbeke 1882-07-24. Side note - Julius and Eulodie/Elodie/Elodia appear to have married in Harelbeke 1880-09-09 with Julius and Theophilus being brothers.
Her parents are cited as Edouardus Coucke age 59 and Rosalia Van Den Broucke (no age given), both living in Harelbeke.
If this is the person you are looking for, then you should be able to use the online records at FamilySearch.org and/or the State Archives of Belgium to find/confirm births, marriages, and deaths of Rene's parents, grandparents, and perhaps more.
If you go back prior to 1796, then the State Archives of Belgium have parochial records online going back into the 1600s.
You may already be well aware of this, but keep in mind that spellings may vary over time and cited ages/dates may not always be accurate.
Hope this helps. Good luck with you research. If you have any questions, feel free to direct message me or email me at firstname.lastname@example.org