sambo - breastfeeding does not prevent eczema. The research on breastfeeding an atopic conditions is very uncertain, and IMO eczema reduction/prevention should not be on lists of supposed benefits.
Breastfeeding is far, far more than this, anyway, as I'm sure you know. Formula feeding has measurable risks to health. As for the sleeping, breastfeeding supports the normal, physiological way for infants to sleep - which is indeed to wake and feed briefly in the night. Sleeping between 11pm and 8pm at a month old is highly unusual in both bf and ff babies, and may well not last.
There are ways to have easier nights for you, now your baby is a bit older, and you could explore these, to address the concerns of your partner.
But ff is not a guarenteed way to get more sleep at all - in fact the research we do have shows that parents get more sleep if they fully bf, as opposed to giving a bottle at night:
Breast-feeding Increases Sleep Duration of New Parents.
Feature Article
Journal of Perinatal & Neonatal Nursing. 21(3):200-206, July/September 2007.
Doan, Therese RN, IBCLC; Gardiner, Annelise; Gay, Caryl L.; Lee, Kathryn A. PhD, RN, FAAN
Abstract:
Objectives: This study describes sleep patterns for mothers and fathers after the birth of their first child and compares exclusive breast-feeding families with parents who used supplementation during the evening or night at 3 months postpartum.
Methods: As part of a randomized clinical trial, the study utilized infant feeding and sleep data at 3 months postpartum from 133 new mothers and fathers. Infant feeding type (breast milk or formula) was determined from parent diaries. Sleep was measured objectively using wrist actigraphy and subjectively using diaries. Lee's General Sleep Disturbance Scale was used to estimate perceived sleep disturbance.
Results: Parents of infants who were breastfed in the evening and/or at night slept an average of 40-45 minutes more than parents of infants given formula. Parents of infants given formula at night also self-reported more sleep disturbance than parents of infants who were exclusively breast-fed at night.
Conclusions: Parents who supplement their infant feeding with formula under the impression that they will get more sleep should be encouraged to continue breast-feeding because sleep loss of more than 30 minutes each night can begin to affect daytime functioning, particularly in those parents who return to work.
Hope this helps