Whether you can safely divide in fall versus spring depends on your climate. It's risky here in USDA Zone 4 but not necessarily so in milder winter areas.
Re nutrients, a high soil pH affects the availability of most micronutrients. Daylilies seem somewhat prone to manganese deficiency (similar symptoms to iron chlorosis) and manganese deficiency in other plants can negatively affect fertility. For example grain yield of corn (maize) is depressed while growth is less so, fewer pollen grains are produced and pollen germination rates are low. Also in corn, zinc deficiency (another micronutrient) decreases pollen viability. In cereal grains, deficiency of another micronutrient, copper, causes inhibition of anther formation, smaller number of pollen grains produced per anther, and non-viability of pollen (partly due to lack of carbohydrate supply to developing pollen). Another, boron, is essential for pollen tube growth - low boron inhibits flowering and development of seed, and may result in seeds with low boron content. Low boron content seeds have low germination rate and a high percentage of resulting seedlings may be abnormal. This info is from the book "Mineral Nutrition of Higher Plants", Horst Marschner, Second Edition, 2002, Academic Press. So to answer the question, a nutrient deficiency caused by high pH can cause low fertility but whether that's the problem in this case I don't know. Stella produces lots of "bee pods" here.
It may be worth considering that amending with manure won't necessarily lower the soil pH.
