Yes, "is is" is grammatically correct in that sentence, and yes it has the exact meaning you reduce it to.

The subject is, "What I think the purpose of life is" (not "<del>**the**</del> life"). This is a noun phrase. The main verb is "is". The subject complement is "to live happily with our loved ones".

Here's some more examples with video clips:

>[The only thing there is is cognition][1]
>
>[What it is is the National Park Service is providing staff...][2]
>
>[And why that is is still a mystery][3]


  [1]: https://youtu.be/qc6ctN5-xX0?t=410
  [2]: https://youtu.be/qM0uOAqrVb0?t=415
  [3]: https://youtu.be/k9ikJ-cuKZ0?t=190