The three most divisive issues in the LP are abortion, immigration, and foreign intervention. What they have in common is that they are all issues of franchise -- issues which deal with an entity's ethical status, based on attributes such as property ownership, religion, race, gender, citizenship, age, intelligence, sentience, and sexual orientation. Issues of enfranchisement lie outside the two-dimensional plane of the Nolan Chart, which is defined by freedom versus security for fully-franchised entities on civil and economic matters. Libertarianism is defined essentially as northward in the Nolan plane, and so gives no guidance on what entities qualify to have their liberty protected. Traditionally, libertarians and other liberals lean toward inclusiveness on franchise issues, but on the Big Three LP Schisms they lean toward inclusiveness only on immigration.
The way that LP Platforms have handled the abortion issue over the decades is instructive. The original 1972 Libertarian Platform only opposed abortion restrictions in the first 100 days of pregnancy, in a discussion of birth control in the "overpopulation" plank. By 1990 there was an abortion plank saying "we support the right of women to make a personal choice regarding the termination of pregnancy". In 1996 the abortion plank lost this language, and replaced it with the "government out of the question / good-faith views" formula and vaguer opposition to "legislation restricting or subsidizing women's access to abortion or other reproductive health services". The latter language was dropped in 2000, and so from 2000 through 2004 the abortion plank did not explicitly say that it opposes abortion bans motivated by what it calls the "good-faith view [...] that abortion is murder". It vaguely said that "government should be kept out of the question", but it only opposed abortion subsidies and mandates. It did not explicitly list termination of pregnancy among its list of the rights of pregnant women. When it called for "repeal of all laws discriminating against women", it didn't list abortion bans among its examples.
The population plank (renamed from "overpopulation") had since at least 1990 changed the 1972 first-100-days language to: "we oppose government actions that either compel or prohibit abortion, sterilization, or any other forms of birth control". The "birth control" proviso could be interpreted not to include late-term abortions, since it evolved from such language and was in the population plank at a time that the abortion plank did not explicitly oppose abortion bans. In 2006 we consolidated the population and abortion planks into a "reproductive rights" plank, and thus without discussion or apparent awareness re-introduced into our abortion discussion what it had lacked for six years: explicit opposition to (only some?) abortion bans.
Thus the original LP Platform opposed only first-trimester abortion bans, and since 1996 the Platform has been much clearer and consistent in admitting that "libertarians can hold good-faith views [...] that abortion is murder" than it has in proposing to eliminate abortion bans. I think it would be wise to apply the "good-faith views" approach to the other two franchise schisms (libervention and immigration), and not commit the LP to any of the six policy extrema that are available on these three issues.