as for your problem if it is SI joint, then it is likely that your position on the bike while hunched over to the handles is causing lumbar spine flexion (and probably t-spine flexion too).  

it is hard to maintain neutral spine position while cycling. some people just put up with it. but like it or not, it encourages flexion and problems like yours.

you may want to tip the seat forward just a little bit to let the pelvis rotate forward with your back. a flat seat may feel more comfortable, but it doesn’t encourage pelvis rotation to help the problem. raising the handle bars and riding on the hoods will help a lot, but you may enjoy that low handlebar/high seat setup.  still, practicing neutral spine position AND learning how to maintain it while riding will be crucial IMHO. 

any MWODS that help with hip flexion will also be good for you. this is my favorite with the band:

this recent DailyRx for achieving a pistol is relevant too:

raising your ability to achieve hip flexion (like for pistol) and maintaining neutral spine will be a big help in alleviating your SI problem. you will be able to pedal, raise your leg to your chest and not push your pelvis up in reaction.