The regular Flybus connection from KEF to Reykjavik stops at the BL. I enjoyed a great half day tour of the Reykjanes Peninsular from Reykjavik ... and then stopped at the BL in the afternoon. Note that the BL is a timed ticket attraction and slots sell out fast. Also there seem to be many unhappy car hire customers from both the majors and local car rental agencies on the Trip Advisor forum, so you may find that your flexibility and 'open road' wishes cost you dearly in 'dings' and stress in waiting times to collect and return the car. Also ... the time of year means that even if you are used to driving in the worst that Chicago can offer you, you may still find Iceland a 'gear too high'! Plus you can get quite a small group tour, and learn lots from your driver/guide which really helps make the visit stand out.
Personally speaking I'd go guided tour all the way ... and if you book say two, try to get a discount, but that's just my personal opinion.