Not sure what you mean by "budget rental". If you mean "Budget", the rental company, you'll have plenty of time (assuming you land on time) to get to the Blue Lagoon. If you mean "budget" as in "cheap" do know that many of the cheap (and "SAD") rental companies rent out crap cars and you are likely to have issues getting to the Blue Lagoon.

That being said, BL to Jokulsarlon is a drive of over 6 hours in best conditions, plus stops for meals, bathrooms, photos (lots of photos) and gas, on narrow and winding roads through mountain passes without guardrails, and should only be attempted on solid rest, not after an overnight international flight and the Blue Lagoon, which will only relax you more and make you less alert, even after that shower.