I suspect you will not find any type of service entrance cable smaller than 6 ga. Far larger than you need for your power requirement for lighting at the 45 foot distance.
It sounds like you want to tie into an existing 20 Amp GFCI receptacle in your garage. Doing this with any type of entrance cable will not be a low cost, practical solution, and will require transition boxes.
If all you need is lighting, it may be more practical and lower cost to consider a solar panel and battery with some inside LED lighting, since this is just for a shed.
You did say that underground is not possible but if it could be done, you could use UF Romex, 12 ga.