Is completing the prayer while trying to hold back gases and reducing the bowing (ruku') to avoid pressure considered a mistake that necessitates repeating the prayer, especially if going to the restroom might lead to missing the time for Fajr prayer?

If you bow for Rukoo' such that your hand touches your knee, the prayer is valid and does not need to be repeated. Otherwise, it must be repeated. Prayer while suppressing the urge to pass wind is valid if one fears the time for the prayer will expire. One must beware of obsessive thoughts (waswas) and turn away from them.