Xjyuk

I am part of a group of immortals from the planet Ziest. From the dawn of time we came…moving silently down through the centuries. Living many secret lives, struggling to reach the time of the gathering, when the few who remain will battle to the last.

We immortals unfortunately cannot create progeny, as our bodies cannot produce the sex cells necessary to make offspring. However, recent technology have allowed immortals to become surrogates, allowing them to pass on their immortal genes to the next generation. A fertilized egg from two mortals is put into the womb of a female immortal, allowing it to gestate there.

During pregnancy, placenta separates the baby’s and the mother’s DNA. When the baby forms in utero, the placenta forms along with it. The primary purpose of the placenta is to be a gatekeeper. It provides the growing baby with nourishment and sustenance, and is one of the main protectors of the integrity of the fetus. Only specific matter can go through to the baby, meaning that the DNA of the carrier stay on the other side of the placenta. This is meant to protect both parties, as too many cells passing between mother and child can be dangerous to both.

How can this be modified to allow the surrogate to pass on their genes without endangering the lives of the carrier or child?

You could change the premise of infertility in the immortals of Zeist from an inability to produce sex cells, to the cells being produced being dud/defective due to an inherited disease of the nuclei.

Once you establish the above, you can then use the technique of three parent baby (used in real life) to allow for transfer of the nuclei in the reproductive cell, which will ensure that the mitochondrial DNA of the immortals of Zeist gets transferred to the babies.

Next, you need to establish that the mitochondrial DNA is source for some of the pecular traits of the immortals of Zeist, which can now be inherited with advances of technology.

The children are chimeras.

In our world, cells from a fetus can move thru the placenta and take up long term residence in the mother.

https://www.scientificamerican.com/article/scientists-discover-childrens-cells-living-in-mothers-brain/

The physical connection between mother and fetus is provided by the
placenta, an organ, built of cells from both the mother and fetus,
which serves as a conduit for the exchange of nutrients, gasses, and
wastes. Cells may migrate through the placenta between the mother and
the fetus, taking up residence in many organs of the body including
the lung, thyroid, muscle, liver, heart, kidney and skin. These may
have a broad range of impacts, from tissue repair and cancer
prevention to sparking immune disorders.

It is remarkable that it is so common for cells from one individual to
integrate into the tissues of another distinct person. We are
accustomed to thinking of ourselves as singular autonomous
individuals, and these foreign cells seem to belie that notion, and
suggest that most people carry remnants of other individuals...

It is easy to detect male cells in a female because you can look for cells with a Y chromosome. There is no reason to doubt that the exchange is reciprocal - mother cells moving in and taking up residence in the fetus.

This is how your immortals pass on their genes. Their cells move thru the placenta and take up residence in the growing fetus. Immortal cells in residence then gradually and gently compete for space with the cells of the native human, taking advantage of the scaffolding grown by the fetus and gradually edging out the natives over time. It might take a lot of time but eventually the individual is effectively a clone of the "surrogate", with the mortal cells of the original having died out over the decades.