This started out as a comment to the image the OP referred to in another answer.
Looking on Google the end of your proposed taxi ride is apparently called
Strandbad Langener Waldsee. If you google this name you will find lots of good reviews on places like Trip Advisor saying that it is a nice, clean place to spend some time swimming. Summarizing what I saw from various reviews:
- Good, clean family friendly location
- Had an entrance fee €3 per person per day in 2015
- There is a bar/snackbar
- Clothing optional was mentioned in some older reviews
- The cement works on the lake shore detract from the view
- There doesn't seem to be much shade
- One person said if you get there in the morning you will have the place to yourself
I was not able to determine if you are able to rent towels or beach umbrellas.
What I would caution is spending 1/2 day in the sun at the beach before a long flight.
Sunburnt children on a plane sounds like the plot of a horror story!
On the other hand, sleeping under a tree/umbrella before your flight sounds like a great way to relax.