if you dont have go out for hours at the time , you can carry your epipen in your back pocket, as long as the temperature is not below 59 degrees or above 86.

if you are going to be out for a long period of time and know you have a life threatning allergie to bee stink, you should definitly carry it although you may not like to carry it, it shouldnt even be a choice and for that, you can find some sort of little case ( made of material) you can store crushed ice in a plastic baggie on one side of the wallet and your epipen on the other, that way your epipen wont overheat and it wont get too cold since the ice would be on the other side of the wallet and it should fit in your back pocket or the bag attached to your bike.

but of course if it's for hours at the time, you need to find a way to refill the ice! but a stop at a store to keep that epipen working is worth it, since we're talking about your life here.
minor adjustments, but it's duable