This is a big complex issue, which in my opinion mostly boils down to the fact that the hypothesis that immortality will cause overpopulation shouldn't be privileged over the hypothesis that it will lead to better population control. Some points to consider:

• Wanting to have kids (to the extent of being willing to cause environmental catastrophe by so doing) is plausibly in part related to the unfulfilled desire to live longer. Reproduction allows one to have an impact on human society which is meaningful over long time scales. Indefinite life extension fulfills this need without increasing the number of individuals.
• Having more than one surviving child per adult on average causes exponential growth problems even if immortality goes unsolved. The solution to this is birth control. The automated natural death cycle, aside from its inhumane aspects, is also arguably a fake solution that deludes people into thinking that contraception and conscious family planning is unnecessary.
• Humans die of accidents and suicide every so often to begin with. Thus there will be new space made for a small number of families to have new children. It will simply not be a major part of the overall culture.
• Space is big enough to mean there is a lot of time to solve the problem if we limit ourselves to natural biological reproduction. Biotech that allows gigantic family sizes with little cost to the parents (e.g. food factories and artificial wombs) is not the result of immortality but of general advancements in biotechnology.
• An engineered lifecycle wherein an individual returns to childhood after a period of elderhood is a plausible substitute for the existing status quo of all children being newly created as blank slates. Children could thus exist, but with a rich set of memories and experiences to draw upon and integrate into their lives.
• The desire to reproduce is largely a cultural artifact, as opposed to an innate drive. To the extent that it is an innate drive, it may be fulfilled by raising any kind of infant-like creature -- including reborn adults, simulated children, and animals.
• Sometimes it is acceptable to focus on near-term problems at the expense of the extremely long-term. All forms of life-saving human technology arguably contribute to human overpopulation, but given the severity of death it is a worthwhile expenditure. We would rather have a large population than have millions die of hunger or disease -- why should aging be different?
• Individuals moving to a silicon substrate would most likely not consume anywhere near the resources that natural humans do.
• Individuals could also choose to move their brain and central nervous system to a low energy-cost nourishment tank, and use robotic bodies that cost far less energy than a natural human body.
• Groups could plausibly merge into single individuals, by forming hive-minds, effectively reducing the total population.