Good thought with the estrogen (in conjunction with progesterone if the woman has her uterus)- that is definitely a potential treatment for osteoporosis, but is generally used only when HRT is needed for other reasons, because there are more effective treatments that carry less risk. There is another hormone that is more frequently used. You are soooooooooooooooooooo close with your thoughts about serum calcium. SO SO close. Follow that train of thought a little farther into the physiology, and you've got the answer.
Off topic, but the midwife in me feels compelled to point out for any readers that the risks of HRT have been widely overblown. For healthy women with vasomotor menopause symptoms (or GU sx that don't respond to local estrogen) and without contraindications, HRT is a safe option with important potential health benefits. The data does not suggest an increased risk of all-cause mortality with HRT use, despite the common public perception.
In fact, despite the truth that HRT does increase the relative risk of breast cancer and multiple CV problems, the data suggests a clinically significant decrease
in all cause mortality.
Image from uptodate, original source is Stuenkel CA, Davis SR, Gompel A, et al. Treatment of Symptoms of the Menopause: An Endocrine Society Clinical Practice Guideline. J Clin Endocrinol Metab 2015.