I've tried to look this up but can't really find much about it, but this is a question that has been on my mind in recent times.
Basically, it is clear to me that Shinto and Buddhism coexisted peacefully, with many of their core beliefs lining up. One conflicting belief, though, stands out to me. In Buddhism, the cycle of rebirth is a central belief, and the focus lies on future lives as much as the current one. In Shinto, it is believed that deceased ancestors would pass on into the other world and be able to visit the present world and protect their descendants.
To be this seems like a rather big contrast. One religion focuses on the present life as well as the future life after rebirth, whereas the other one honors their dead for many generations.
How did these two religions manage to coexist peacefully while their core beliefs about the afterlife didn't seem to line up? Or am I missing something about their perspectives on the afterlife that does make them work well together? Or did the Japanese people simply 'pick' one of the two beliefs?
I do not know the intricacies of the afterlife that the pre-modern Japanese believed in, but I can tell you about how these two religions "coexisted". The short answer is that they were never two separate religions before the Meiji restoration at all.
Teeuwen (2004) writes that "...within a few years after the coup [Meiji restoration] in January 1868 the new government started a "purification" of holy sites throughout the entire country. Priests with a Buddhist background were replaced, gods with Buddhist names and titles had to get back their "original" identity and Buddhist buildings en items had to be destroyed or removed. The slogan under which these measures were carried out was shinbutsu bunri: separation of Shinto and Buddhism." (original text in Dutch)
What is interesting about this is that there had to be a separation process at all, and a nation-wide one at that. This means that before, the two religions were fused, as it were. The aforementioned destruction or removal of Buddhist items or buildings is talking about holy sites that are "shared" by Shinto and Buddhism. But, perhaps the strongest evidence is that Buddhist priests could also hold positions in Shinto shrines. After all, if a single person can serve these two, apparently separate religions in a serious manner as a priest, then those religions cannot truly be all that separate. And if they are not separate religions but rather one, fused religion, then there is not really a question of coexistence.
Sources:
Andrew Gordon, A Modern History of Japan, Oxford University Press, 2009.
Mark Teeuwen, Shinto, Amsterdam University Press, 2004.
It seems a bit odd to me that you're asking this about Feudal Japan. By that time, Buddhism and Shinto had co-existed for centuries, and any theological/philosophical differences between them had largely been "solved", in a sense.
When Buddhism first spread to Japan from China (in the mid-6th century) it was already syncretised with many aspects of Confucianism, and during its introduction in Japan it was further syncretised with the beliefs and practices already present in Japan (the Japanese "folk religion", as some call it). So Mahayana Buddhist beliefs were altered to fit in with existing practices and beliefs.
Early Shinto (or perhaps proto-Shinto would be more accurate?) didn't have a lot to say about the afterlife, and as Shinto developed into a more popular and more complex religion it largely incorporated the same indigenous beliefs and practices about the afterlife that Japanese Buddhism had also syncretised.
I'm not an expert on actual Japanese religious beliefs/tenets, modern or historical, but the beliefs resulting from these merged religions was, for example (and this is overly simplistic, to be sure), a continued belief in the tenets of Mahayana Buddhism, including reincarnation... but indigenous Japanese funerary and ancestor worship practices were also important in order to help the deceased progress along Bodhisattvayāna, and so that your great ancestors in Samsara can gift you with good luck, etc.
So getting back on track -- when someone died in Feudal Japan the funeral, memorial services, tomb maintenance, shrine visits, anniversary rites, etc, were a mixture of practices. Some of those practices/rituals originally came from Buddhism, some came from Shinto, and some came from traditional indigenous practices. But these mixed practices had all been ordinary and accepted for centuries. Likewise, the beliefs and perspectives of the religions themselves had been melded for centuries. So why would any theological conflicts arise over it? These mixed practices and beliefs were "normal", or even "traditional" to any Japanese person in the Feudal period (unlike those newfangled Christian missionaries...).
If you want to read more about perspectives on ancestor worship within early Japanese Buddhism, I'll suggest checking out Choon Sup Bae's Ancestor worship and the challenges it poses to the Christian mission and ministry. As the name indicates, the work is largely about Christian issues in Japan, but chapter 5 has a very good introduction to ancestor worship in Japan, including its early adoption into both Japanese Buddhism and [proto-]Shinto.
As for sources discussing Feudal Japan, there are plentiful sources about religion in Japan during those times, but I couldn't find anything specifically about Buddhist-vs-Shinto perspectives or conflicts, because they are so intertwined and harmonized by then.