Yes, the example sentence is grammatically correct, even with "is is", and yes it has the meaning you reduce it to.

The subject is, "What I think the purpose of life is". 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

What it is is the National Park Service is providing staff...

And why that is is still a mystery

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 "the 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

What it is is the National Park Service is providing staff...

And why that is is still a mystery