Immortality + reproduction means population rate expands at a blistering pace. You cannot build generation ships big enough, and you will fill up every habitable scrap of the solar system, including artificial habitats.

If there is no upper limit on population growth, you also need no upper limit on your rate of expansion.

Suspended animation could work, but now you've just admitted that you're putting a limit on reproduction, but even that ends when those people wake up and arrive somewhere.

Even allowing all that, "we'll solve this unsolved problem by solving another unsolved problem!" does not inspire a lot of confidence.

You're running afoul of the pitfalls of static analysis. Humans don't have an immutable reproduction rate of X% every Y years.

Immortality will reduce reproduction rates intrinsically; 500 years ago it was common to be married off and breeding before you were 15, now you're 'having kids young' if you start before 25, how long will people wait when they live 100s of years?.

Improving global standards of education and quality of living also slow down reproduction. Countries scoring highest in both of these metrics are often fairly stable in population, some even depend on immigration to maintain current levels.

Residual growth after these two factors are taken into account is low enough that the odd colonisation attempt or newly built orbital habitat will be ample space. And this is without even considering that we could end up living as uploads in virtual worlds long before these things become possible.

That doesn't solve the problem. All you do is move the problem beyond our star.

Entropy is still increasing.