I used to be a Regional Archery Judge in the GWAS region. I have never transported my bow by bike... partly due to distances and early competition starts and partly due to the fact that my bow is approx £1200 and my arrows £200 and I wouldn't risk them being damaged. I did know someone who would cycle up to 50miles to competition. Their kit would be spread throughout panniers and back pack, but it wasn't as valuable as mine.
I have also just thought that on a bike I wouldn't want anything vibrating loose and affecting the tuning of the bow, such as tiller adjustment and pressure button settings, as on a road bike it would all get shaken around a fair bit.
Hugs
Archeress x
PS I shoot recurve.